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Theorem prprc 4663
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 4661 . 2 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
2 snprc 4613 . . 3 𝐵 ∈ V ↔ {𝐵} = ∅)
32biimpi 219 . 2 𝐵 ∈ V → {𝐵} = ∅)
41, 3sylan9eq 2853 1 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2111  Vcvv 3441  c0 4243  {csn 4525  {cpr 4527
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-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-pr 4528
This theorem is referenced by: (None)
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