Theorem opelxpii 5657

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 5656	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
4	1, 2, 3	mp2an 692	1 ⊢ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)

Syntax hints:   ∈ wcel 2113  ⟨cop 4581   × cxp 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-opab 5156  df-xp 5625

This theorem is referenced by:  pzriprnglem7  21426  pzriprng1ALT  21435  grlimedgnedg  48255