Theorem opprc1 3759

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 3758. (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

Step	Hyp	Ref	Expression
1		simpl 108	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
2	1	con3i 622	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		opprc 3758	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
4	2, 3	syl 14	1 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 2125  Vcvv 2709  ∅c0 3390  ⟨cop 3559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-dif 3100  df-nul 3391  df-op 3565

This theorem is referenced by:  brprcneu  5454