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Theorem prprc 4767
Description: An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
prprc ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)

Proof of Theorem prprc
StepHypRef Expression
1 prprc1 4765 . 2 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
2 snprc 4717 . . 3 𝐵 ∈ V ↔ {𝐵} = ∅)
32biimpi 215 . 2 𝐵 ∈ V → {𝐵} = ∅)
41, 3sylan9eq 2788 1 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1534  wcel 2099  Vcvv 3470  c0 4318  {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 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-dif 3948  df-un 3950  df-nul 4319  df-sn 4625  df-pr 4627
This theorem is referenced by: (None)
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