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Theorem opprc1 4789
Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4788. (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 486 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
2 opprc 4788 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
31, 2nsyl5 162 1 𝐴 ∈ 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  cop 4531
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-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-op 4532
This theorem is referenced by:  snopeqop  5361  epelg  5431  brprcneu  6637  fmlafvel  32742  bj-inftyexpidisj  34622  eu2ndop1stv  43676
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