Theorem brv 5458

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 5115	. 2 ⊢ (𝐴V𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ V)
3	1, 2	mpbir 234	1 ⊢ 𝐴V𝐵

Syntax hints:   ∈ wcel 2150  Vcvv 3462  ⟨cop 4600   class class class wbr 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-un 3918  df-in 3920  df-ss 3930  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115

This theorem is referenced by:  brsset  36337  brtxpsd  36342  dffun10  36362  elfuns  36363  dfint3  36402  brub  36404  brvdif  38865