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Theorem opelxpii 40591
Description: Ordered pair membership in a Cartesian product (implication). (Contributed by Steven Nguyen, 17-Jul-2022.)
Hypotheses
Ref Expression
opelxpii.1 𝐴𝐶
opelxpii.2 𝐵𝐷
Assertion
Ref Expression
opelxpii 𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)

Proof of Theorem opelxpii
StepHypRef Expression
1 opelxpii.1 . 2 𝐴𝐶
2 opelxpii.2 . 2 𝐵𝐷
3 opelxpi 5668 . 2 ((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
41, 2, 3mp2an 690 1 𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  cop 4590   × cxp 5629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5166  df-xp 5637
This theorem is referenced by: (None)
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