Theorem brprc 2097

Description: A property of proper class as the second argument of a binary relation.

Assertion

Ref	Expression
brprc	⊢ (¬ B ∈ V → (ARB ↔ ARA))

Proof of Theorem brprc

Step	Hyp	Ref	Expression
1		opprc2 1907	. . 3 ⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)
2	1	eleq1d 1155	. 2 ⊢ (¬ B ∈ V → (⟨A, B⟩ ∈ R ↔ ⟨A, A⟩ ∈ R))
3		df-br 2063	. 2 ⊢ (ARB ↔ ⟨A, B⟩ ∈ R)
4		df-br 2063	. 2 ⊢ (ARA ↔ ⟨A, A⟩ ∈ R)
5	2, 3, 4	3bitr4g 428	1 ⊢ (¬ B ∈ V → (ARB ↔ ARA))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054

This theorem is referenced by:  vtoclrbr 2450  vtoclibr 2451  f1oeng 3298  f1domg 3299  sdomex 3315  unen 3338  numth2 3600  cardval 3633

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063

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