Theorem xpss1 5573

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 3988	. 2 ⊢ 𝐶 ⊆ 𝐶
2		xpss12 5569	. 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 3935   × cxp 5552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-in 3942  df-ss 3951  df-opab 5128  df-xp 5560

This theorem is referenced by:  ssres2  5880  funssxp  6534  tposssxp  7895  tpostpos2  7912  unxpwdom2  9051  dfac12lem2  9569  unctb  9626  axdc3lem  9871  fpwwe2  10064  pwfseqlem5  10084  wrdexgOLD  13871  imasvscafn  16809  imasvscaf  16811  gasubg  18431  mamures  21000  mdetrlin  21210  mdetrsca  21211  mdetunilem9  21228  mdetmul  21231  tx1cn  22216  cxpcn3  25328  imadifxp  30350  1stmbfm  31518  sxbrsigalem0  31529  cvmlift2lem1  32549  cvmlift2lem9  32558  poimirlem32  34923  trclexi  39978  cnvtrcl0  39984  volicoff  42279  volicofmpt  42281  issmflem  43003