Theorem brv 5457

Description: Two classes are always in relation by V. This is simply equivalent to ⟨𝐴, 𝐵⟩ ∈ V, and does not imply that V is a relation: see nrelv 5790. (Contributed by Scott Fenton, 11-Apr-2012.)

Assertion

Ref	Expression
brv	⊢ 𝐴V𝐵

Proof of Theorem brv

Step	Hyp	Ref	Expression
1		opex 5449	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		df-br 5124	. 2 ⊢ (𝐴V𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ V)
3	1, 2	mpbir 231	1 ⊢ 𝐴V𝐵

Syntax hints:   ∈ wcel 2107  Vcvv 3463  ⟨cop 4612   class class class wbr 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124

This theorem is referenced by:  brsset  35849  brtxpsd  35854  dffun10  35874  elfuns  35875  dfint3  35912  brub  35914  brvdif  38221