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Theorem opprc2 3600
Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 3598. (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 107 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
21con3i 572 . 2 𝐵 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 opprc 3598 . 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 101   = wceq 1259  wcel 1409  Vcvv 2574  c0 3252  cop 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-nul 3253  df-op 3412
This theorem is referenced by: (None)
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