Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpss1 GIF version

Theorem xpss1 4649
 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 3117 . 2 𝐶𝐶
2 xpss12 4646 . 2 ((𝐴𝐵𝐶𝐶) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
31, 2mpan2 421 1 (𝐴𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ⊆ wss 3071   × 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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084  df-opab 3990  df-xp 4545 This theorem is referenced by:  ssres2  4846  ssxp1  4975  funssxp  5292  tposssxp  6146  tpostpos2  6162  tfrlemibfn  6225  tfr1onlembfn  6241  tfrcllembfn  6254  enq0enq  7246  tx1cn  12448
 Copyright terms: Public domain W3C validator