Theorem opex 3966

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 3964	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 402	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ V

Syntax hints:   ∈ wcel 1393  Vcvv 2557  ⟨cop 3378

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384

This theorem is referenced by:  opabid  3994  elopab  3995  opabm  4017  elvvv  4403  xpiindim  4473  raliunxp  4477  rexiunxp  4478  intirr  4711  xpmlem  4744  dmsnm  4786  dmsnopg  4792  cnvcnvsn  4797  cnviinm  4859  funopg  4934  fsn  5335  idref  5396  oprabid  5537  dfoprab2  5552  rnoprab  5587  fo1st  5784  fo2nd  5785  eloprabi  5822  xporderlem  5852  dmtpos  5871  rntpos  5872  tpostpos  5879  iinerm  6178  th3qlem2  6209  ensn1  6276  xpsnen  6295  xpcomco  6300  xpassen  6304  phplem2  6316  ac6sfi  6352  genipdm  6614  ioof  8840