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Theorem prnesn 4856
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 2753 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
21necon3ai 2960 . . 3 (𝐴𝐵 → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
323ad2ant3 1133 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
4 simp1 1134 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐴𝑉)
5 simp2 1135 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐵𝑊)
64, 5preqsnd 4855 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
76necon3abid 2972 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ≠ {𝐶} ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐶)))
83, 7mpbird 257 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ≠ {𝐶})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2935  {csn 4624  {cpr 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-v 3471  df-dif 3947  df-un 3949  df-nul 4319  df-sn 4625  df-pr 4627
This theorem is referenced by:  prneprprc  4857  snnen2o  9255
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