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Theorem prnesn 4664
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 2801 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
21necon3ai 2992 . . 3 (𝐴𝐵 → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
323ad2ant3 1115 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
4 elex 3433 . . . . 5 (𝐴𝑉𝐴 ∈ V)
543ad2ant1 1113 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐴 ∈ V)
6 elex 3433 . . . . 5 (𝐵𝑊𝐵 ∈ V)
763ad2ant2 1114 . . . 4 ((𝐴𝑉𝐵𝑊𝐴𝐵) → 𝐵 ∈ V)
85, 7preqsnd 4663 . . 3 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))
98necon3abid 3003 . 2 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ≠ {𝐶} ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐶)))
103, 9mpbird 249 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ≠ {𝐶})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2967  Vcvv 3415  {csn 4441  {cpr 4443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-v 3417  df-dif 3832  df-un 3834  df-nul 4179  df-sn 4442  df-pr 4444
This theorem is referenced by:  prneprprc  4665
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