Theorem opelvv 4714

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 4696	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
4	1, 2, 3	mp2an 426	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)

Syntax hints:   ∈ wcel 2167  Vcvv 2763  ⟨cop 3626   × cxp 4662

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-opab 4096  df-xp 4670

This theorem is referenced by:  relsnop  4770  relopabi  4792  eqop2  6245