Theorem xpss2 4658

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 3122	. 2 ⊢ 𝐶 ⊆ 𝐶
2		xpss12 4654	. 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 3076   × cxp 4545

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3082  df-ss 3089  df-opab 3998  df-xp 4553

This theorem is referenced by:  ssxp2  4984  xpdom3m  6736  axresscn  7692  tx2cn  12478  dvfvalap  12858