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Theorem opprc2 4566
Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 4564. (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 479 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
21con3i 150 . 2 𝐵 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 opprc 4564 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
42, 3syl 17 1 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1620  wcel 2127  Vcvv 3328  c0 4046  cop 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-v 3330  df-dif 3706  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-op 4316
This theorem is referenced by:  dmsnopss  5754  strle1  16146
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