Theorem opelxpii 5707

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 5706	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
4	1, 2, 3	mp2an 689	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)

Syntax hints:   ∈ wcel 2098  ⟨cop 4629   × cxp 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-xp 5675

This theorem is referenced by:  pzriprnglem7  21374  pzriprng1ALT  21383