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Theorem prprc 4436
 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 4434 . 2 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
2 snprc 4387 . . 3 𝐵 ∈ V ↔ {𝐵} = ∅)
32biimpi 206 . 2 𝐵 ∈ V → {𝐵} = ∅)
41, 3sylan9eq 2824 1 ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144  Vcvv 3349  ∅c0 4061  {csn 4314  {cpr 4316 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-dif 3724  df-un 3726  df-nul 4062  df-sn 4315  df-pr 4317 This theorem is referenced by: (None)
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