Theorem opex 3940

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 3938	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → ⟨A, B⟩ ∈ V)
4	1, 2, 3	mp2an 404	1 ⊢ ⟨A, B⟩ ∈ V

Syntax hints:   ∈ wcel 1374  Vcvv 2535  ⟨cop 3353

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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359

This theorem is referenced by:  opabid  3968  elopab  3969  opabm  3991  elvvv  4330  xpiindim  4400  raliunxp  4404  rexiunxp  4405  intirr  4638  xpmlem  4671  dmsnm  4713  dmsnopg  4719  cnvcnvsn  4724  cnviinm  4786  funopg  4860  fsn  5260  idref  5321  oprabid  5461  dfoprab2  5475  rnoprab  5510  fo1st  5707  fo2nd  5708  eloprabi  5745  xporderlem  5774  dmtpos  5793  rntpos  5794  tpostpos  5801  tfrlemi14  5869  iinerm  6089  th3qlem2  6120  genipdm  6370  genpelpw  6371