Theorem opelxpii 39919

Description: Ordered pair membership in a Cartesian product (implication). (Contributed by Steven Nguyen, 17-Jul-2022.)

Hypotheses

Ref	Expression
opelxpii.1	⊢ 𝐴 ∈ 𝐶
opelxpii.2	⊢ 𝐵 ∈ 𝐷

Assertion

Ref	Expression
opelxpii	⊢ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)

Proof of Theorem opelxpii

Step	Hyp	Ref	Expression
1		opelxpii.1	. 2 ⊢ 𝐴 ∈ 𝐶
2		opelxpii.2	. 2 ⊢ 𝐵 ∈ 𝐷
3		opelxpi 5588	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
4	1, 2, 3	mp2an 692	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)

Syntax hints:   ∈ wcel 2110  ⟨cop 4547   × cxp 5549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-opab 5116  df-xp 5557

This theorem is referenced by: (None)