Theorem xpss1 5719

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

Assertion

Ref	Expression
xpss1	⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))

Proof of Theorem xpss1

Step	Hyp	Ref	Expression
1		ssid 4031	. 2 ⊢ 𝐶 ⊆ 𝐶
2		xpss12 5715	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐶) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
3	1, 2	mpan2 690	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3976   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ss 3993  df-opab 5229  df-xp 5706

This theorem is referenced by:  ssres2  6034  funssxp  6776  tposssxp  8271  tpostpos2  8288  unxpwdom2  9657  dfac12lem2  10214  unctb  10273  axdc3lem  10519  fpwwe2  10712  pwfseqlem5  10732  imasvscafn  17597  imasvscaf  17599  gasubg  19342  mamures  22422  mdetrlin  22629  mdetrsca  22630  mdetunilem9  22647  mdetmul  22650  tx1cn  23638  cxpcn3  26809  imadifxp  32623  1stmbfm  34225  sxbrsigalem0  34236  cvmlift2lem1  35270  cvmlift2lem9  35279  poimirlem32  37612  dfno2  43390  trclexi  43582  cnvtrcl0  43588  volicoff  45916  volicofmpt  45918  issmflem  46648