Theorem opelvvg 5575

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 3418	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3418	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		opelxpi 5572	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
4	1, 2, 3	syl2an 599	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2114  Vcvv 3400  ⟨cop 4532   × cxp 5533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-v 3402  df-dif 3856  df-un 3858  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-opab 5103  df-xp 5541

This theorem is referenced by:  relsnopg  5657  isof1oopb  7104  opvtxfv  26962  opiedgfv  26965  gonafv  32896  sat1el2xp  32925  opelvvdif  36054  brxrn  36160