Theorem xpss2 4694

Description: Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Assertion

Ref	Expression
xpss2	⊢ (𝐴 ⊆ 𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))

Proof of Theorem xpss2

Step	Hyp	Ref	Expression
1		ssid 3148	. 2 ⊢ 𝐶 ⊆ 𝐶
2		xpss12 4690	. 2 ⊢ ((𝐶 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐵) → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))
3	1, 2	mpan 421	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))

Syntax hints:   → wi 4   ⊆ wss 3102   × cxp 4581

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 4589

This theorem is referenced by:  ssxp2  5020  xpdom3m  6772  axresscn  7763  tx2cn  12630  dvfvalap  13010