Theorem opelvv 4676

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 4658	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
4	1, 2, 3	mp2an 426	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)

Syntax hints:   ∈ wcel 2148  Vcvv 2737  ⟨cop 3595   × cxp 4624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4065  df-xp 4632

This theorem is referenced by:  relsnop  4732  relopabi  4752  eqop2  6178