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Theorem xpss1 4696
Description: Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
Assertion
Ref Expression
xpss1 (𝐴𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))

Proof of Theorem xpss1
StepHypRef Expression
1 ssid 3148 . 2 𝐶𝐶
2 xpss12 4693 . 2 ((𝐴𝐵𝐶𝐶) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
31, 2mpan2 422 1 (𝐴𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3102   × cxp 4584
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-in 3108  df-ss 3115  df-opab 4026  df-xp 4592
This theorem is referenced by:  ssres2  4893  ssxp1  5022  funssxp  5339  tposssxp  6196  tpostpos2  6212  tfrlemibfn  6275  tfr1onlembfn  6291  tfrcllembfn  6304  enq0enq  7351  tx1cn  12680
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