Theorem brv 5434

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 5761. (Contributed by Scott Fenton, 11-Apr-2012.)

Assertion

Ref	Expression
brv	⊢ 𝐴V𝐵

Proof of Theorem brv

Step	Hyp	Ref	Expression
1		opex 5426	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		df-br 5111	. 2 ⊢ (𝐴V𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ V)
3	1, 2	mpbir 230	1 ⊢ 𝐴V𝐵

Syntax hints:   ∈ wcel 2106  Vcvv 3446  ⟨cop 4597   class class class wbr 5110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111

This theorem is referenced by:  brsset  34550  brtxpsd  34555  dffun10  34575  elfuns  34576  dfint3  34613  brub  34615  brvdif  36794