Theorem opelxpii 5738

Description: Ordered pair membership in a Cartesian product (implication), induction form. (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 5737	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
4	1, 2, 3	mp2an 691	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)

Syntax hints:   ∈ wcel 2108  ⟨cop 4654   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706

This theorem is referenced by:  pzriprnglem7  21523  pzriprng1ALT  21532