Theorem opex 3918

Description: An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.)

Hypotheses

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

Assertion

Ref	Expression
opex	⊢ ⟨A, B⟩ ∈ V

Proof of Theorem opex

Step	Hyp	Ref	Expression
1		opex.1	. 2 ⊢ A ∈ V
2		opex.2	. 2 ⊢ B ∈ V
3		opexg 3916	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → ⟨A, B⟩ ∈ V)
4	1, 2, 3	mp2an 404	1 ⊢ ⟨A, B⟩ ∈ V

Syntax hints:   ∈ wcel 1375  Vcvv 2533  ⟨cop 3330

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896

This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336

This theorem is referenced by:  opabid  3946  elopab  3947  opabm  3970  elvvv  4296  xpiindim  4366  raliunxp  4370  rexiunxp  4371  intirr  4605  xpmlem  4638  dmsnm  4680  dmsnopg  4686  cnvcnvsn  4691  cnviinm  4753  funopg  4827  fsn  5227  idref  5288  oprabid  5430  dfoprab2  5444  rnoprab  5479  fo1st  5673  fo2nd  5674  eloprabi  5711  xporderlem  5740  dmtpos  5759  rntpos  5760  tpostpos  5767  tfrlemi14  5834