Theorem xpss1 5703

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 4005	. 2 ⊢ 𝐶 ⊆ 𝐶
2		xpss12 5699	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐶) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
3	1, 2	mpan2 691	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3950   × cxp 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ss 3967  df-opab 5205  df-xp 5690

This theorem is referenced by:  ssres2  6021  funssxp  6763  tposssxp  8256  tpostpos2  8273  unxpwdom2  9629  dfac12lem2  10186  unctb  10245  axdc3lem  10491  fpwwe2  10684  pwfseqlem5  10704  imasvscafn  17583  imasvscaf  17585  gasubg  19321  mamures  22402  mdetrlin  22609  mdetrsca  22610  mdetunilem9  22627  mdetmul  22630  tx1cn  23618  cxpcn3  26792  imadifxp  32615  1stmbfm  34263  sxbrsigalem0  34274  cvmlift2lem1  35308  cvmlift2lem9  35317  poimirlem32  37660  dfno2  43446  trclexi  43638  cnvtrcl0  43644  volicoff  46015  volicofmpt  46017  issmflem  46747