Theorem opelvvg 5695

Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)

Assertion

Ref	Expression
opelvvg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))

Proof of Theorem opelvvg

Step	Hyp	Ref	Expression
1		elex 3480	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3480	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		opelxpi 5691	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
4	1, 2, 3	syl2an 596	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3459  ⟨cop 4607   × cxp 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-opab 5182  df-xp 5660

This theorem is referenced by:  relsnopg  5782  isof1oopb  7318  opvtxfv  28983  opiedgfv  28986  gonafv  35372  sat1el2xp  35401  opelvvdif  38277  brxrn  38392