Theorem brv 5486

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 5817. (Contributed by Scott Fenton, 11-Apr-2012.)

Assertion

Ref	Expression
brv	⊢ 𝐴V𝐵

Proof of Theorem brv

Step	Hyp	Ref	Expression
1		opex 5478	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		df-br 5152	. 2 ⊢ (𝐴V𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ V)
3	1, 2	mpbir 231	1 ⊢ 𝐴V𝐵

Syntax hints:   ∈ wcel 2108  Vcvv 3481  ⟨cop 4640   class class class wbr 5151

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5152

This theorem is referenced by:  brsset  35884  brtxpsd  35889  dffun10  35909  elfuns  35910  dfint3  35947  brub  35949  brvdif  38257