Theorem brv 5462

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 5454	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		df-br 5139	. 2 ⊢ (𝐴V𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ V)
3	1, 2	mpbir 230	1 ⊢ 𝐴V𝐵

Syntax hints:   ∈ wcel 2098  Vcvv 3466  ⟨cop 4626   class class class wbr 5138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139

This theorem is referenced by:  brsset  35356  brtxpsd  35361  dffun10  35381  elfuns  35382  dfint3  35419  brub  35421  brvdif  37619