Theorem opelvv 4820

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 4801	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
4	1, 2, 3	mp2an 430	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)

Syntax hints:   ∈ wcel 2209  Vcvv 2821  ⟨cop 3708   × cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188  df-xp 4775

This theorem is referenced by:  relsnop  4876  relopabi  4900  eqop2  6402