Theorem opelvv 4769

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 4751	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
4	1, 2, 3	mp2an 426	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)

Syntax hints:   ∈ wcel 2200  Vcvv 2799  ⟨cop 3669   × cxp 4717

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146  df-xp 4725

This theorem is referenced by:  relsnop  4825  relopabi  4847  eqop2  6324