Theorem opelvv 5674

Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
opelvv.1	⊢ 𝐴 ∈ V
opelvv.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
opelvv	⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)

Proof of Theorem opelvv

Step	Hyp	Ref	Expression
1		opelvv.1	. 2 ⊢ 𝐴 ∈ V
2		opelvv.2	. 2 ⊢ 𝐵 ∈ V
3		opelxpi 5671	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
4	1, 2, 3	mp2an 693	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)

Syntax hints:   ∈ wcel 2114  Vcvv 3442  ⟨cop 4588   × cxp 5632

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5640

This theorem is referenced by:  relopabiALT  5782  funsneqopb  7109  isof1oopb  7283  1st2ndb  7985  eqop2  7988  evlfcl  18159  brtxp  36100  brpprod  36105  brsset  36109  brcart  36152  brcup  36159  brcap  36160  elcnvlem  43986  swapfelvv  49651  fucoelvv  49708  prcofelvv  49768