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Theorem opelxpd 4572
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 4571 . 2 ((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
41, 2, 3syl2anc 408 1 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1480  cop 3530   × cxp 4537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545
This theorem is referenced by:  suplocsrlemb  7626  seqvalcd  10244  ctiunctlemfo  11963  strslfv2d  12015  txcnp  12454  upxp  12455  txcnmpt  12456  uptx  12457  txdis1cn  12461  txlm  12462  lmcn2  12463  txhmeo  12502  comet  12682  txmetcnp  12701  dvaddxxbr  12848  dvmulxxbr  12849  dvcoapbr  12854
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