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Theorem opprc1 4828
Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4827. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opprc1 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Proof of Theorem opprc1
StepHypRef Expression
1 simpl 483 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
2 opprc 4827 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
31, 2nsyl5 159 1 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  c0 4256  cop 4567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-nul 4257  df-if 4460  df-op 4568
This theorem is referenced by:  snopeqop  5420  epelg  5496  brprcneu  6764  brprcneuALT  6765  fmlafvel  33347  bj-inftyexpidisj  35381  eu2ndop1stv  44617
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