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Theorem opelxpii 39431
 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 5557 . 2 ((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
41, 2, 3mp2an 691 1 𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2111  ⟨cop 4531   × cxp 5518 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5094  df-xp 5526 This theorem is referenced by: (None)
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