Theorem xpss1 5619

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 3948	. 2 ⊢ 𝐶 ⊆ 𝐶
2		xpss12 5615	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐶) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
3	1, 2	mpan2 689	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3892   × cxp 5598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-in 3899  df-ss 3909  df-opab 5144  df-xp 5606

This theorem is referenced by:  ssres2  5931  funssxp  6659  tposssxp  8077  tpostpos2  8094  unxpwdom2  9395  dfac12lem2  9950  unctb  10011  axdc3lem  10256  fpwwe2  10449  pwfseqlem5  10469  imasvscafn  17297  imasvscaf  17299  gasubg  18957  mamures  21588  mdetrlin  21800  mdetrsca  21801  mdetunilem9  21818  mdetmul  21821  tx1cn  22809  cxpcn3  25950  imadifxp  30989  1stmbfm  32276  sxbrsigalem0  32287  cvmlift2lem1  33313  cvmlift2lem9  33322  poimirlem32  35857  trclexi  41441  cnvtrcl0  41447  volicoff  43765  volicofmpt  43767  issmflem  44495