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Theorem opprc2 4859
Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 4857. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opprc2 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Proof of Theorem opprc2
StepHypRef Expression
1 simpr 486 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
2 opprc 4857 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
31, 2nsyl5 159 1 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3447  c0 4286  cop 4596
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-dif 3917  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-op 4597
This theorem is referenced by:  snopeqop  5467  dmsnopss  6170  strle1  17038
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