Theorem opelxpii 5683

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 5682	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
4	1, 2, 3	mp2an 702	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)

Syntax hints:   ∈ wcel 2141  ⟨cop 4587   × cxp 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-opab 5162  df-xp 5651

This theorem is referenced by:  pzriprnglem7  21519  pzriprng1ALT  21528  grlimedgnedg  48717