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Theorem opprc2 3563
Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 3561. (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 103 . . 3 ((A V B V) → B V)
21con3i 561 . 2 B V → ¬ (A V B V))
3 opprc 3561 . 2 (¬ (A V B V) → ⟨A, B⟩ = ∅)
42, 3syl 14 1 B V → ⟨A, B⟩ = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  c0 3218  cop 3370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-nul 3219  df-op 3376
This theorem is referenced by: (None)
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