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Theorem prnesn 4753
 Description: A proper unordered pair is not a (proper or improper) singleton. (Contributed by AV, 13-Jun-2022.)
Assertion
Ref Expression
prnesn ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ≠ {𝐶})

Proof of Theorem prnesn
StepHypRef Expression
1 eqtr3 2820 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
21necon3ai 3012 . . 3 (𝐴𝐵 → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
323ad2ant3 1132 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
4 elex 3460 . . . . 5 (𝐴𝑉𝐴 ∈ V)
543ad2ant1 1130 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐴 ∈ V)
6 elex 3460 . . . . 5 (𝐵𝑊𝐵 ∈ V)
763ad2ant2 1131 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐵 ∈ V)
85, 7preqsnd 4752 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
98necon3abid 3023 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ≠ {𝐶} ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐶)))
103, 9mpbird 260 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ≠ {𝐶})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3442  {csn 4528  {cpr 4530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3444  df-dif 3886  df-un 3888  df-nul 4247  df-sn 4529  df-pr 4531 This theorem is referenced by:  prneprprc  4754
 Copyright terms: Public domain W3C validator
The text passage is a formal logic proof from the Metamath Proof Explorer, which proves that a proper unordered pair is not a singleton. It does not contain any mathematical questions or answers in the form of word problems or calculations.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  prnesn Structured version   Visualization version   GIF version

Theorem prnesn 4753
 Description: A proper unordered pair is not a (proper or improper) singleton. (Contributed by AV, 13-Jun-2022.)
Assertion
Ref Expression
prnesn ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ≠ {𝐶})

Proof of Theorem prnesn
StepHypRef Expression
1 eqtr3 2820 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
21necon3ai 3012 . . 3 (𝐴𝐵 → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
323ad2ant3 1132 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
4 elex 3460 . . . . 5 (𝐴𝑉𝐴 ∈ V)
543ad2ant1 1130 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐴 ∈ V)
6 elex 3460 . . . . 5 (𝐵𝑊𝐵 ∈ V)
763ad2ant2 1131 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐵 ∈ V)
85, 7preqsnd 4752 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
98necon3abid 3023 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ≠ {𝐶} ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐶)))
103, 9mpbird 260 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ≠ {𝐶})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3442  {csn 4528  {cpr 4530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3444  df-dif 3886  df-un 3888  df-nul 4247  df-sn 4529  df-pr 4531 This theorem is referenced by:  prneprprc  4754
 Copyright terms: Public domain W3C validator
user
The text passage is a formal logic proof from the Metamath Proof Explorer, which proves that a proper unordered pair is not a singleton. It does not contain any mathematical questions or answers in the form of word problems or calculations.

TYPE: NO_QUESTION