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Theorem opelxpd 4500
Description: Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
opelxpd.1 (𝜑𝐴𝐶)
opelxpd.2 (𝜑𝐵𝐷)
Assertion
Ref Expression
opelxpd (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))

Proof of Theorem opelxpd
StepHypRef Expression
1 opelxpd.1 . 2 (𝜑𝐴𝐶)
2 opelxpd.2 . 2 (𝜑𝐵𝐷)
3 opelxpi 4499 . 2 ((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
41, 2, 3syl2anc 404 1 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1445  cop 3469   × cxp 4465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-opab 3922  df-xp 4473
This theorem is referenced by:  strslfv2d  11685  comet  12285
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