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Theorem opelxpd 4660
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 4659 . 2 ((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
41, 2, 3syl2anc 411 1 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  cop 3596   × cxp 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-opab 4066  df-xp 4633
This theorem is referenced by:  suplocsrlemb  7805  seqvalcd  10459  ctiunctlemfo  12440  strslfv2d  12505  imasaddfnlemg  12735  imasaddflemg  12737  txcnp  13774  upxp  13775  txcnmpt  13776  uptx  13777  txdis1cn  13781  txlm  13782  lmcn2  13783  txhmeo  13822  comet  14002  txmetcnp  14021  dvaddxxbr  14168  dvmulxxbr  14169  dvcoapbr  14174
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