Theorem xpss 5668

Description: A Cartesian product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Assertion

Ref	Expression
xpss	⊢ (𝐴 × 𝐵) ⊆ (V × V)

Proof of Theorem xpss

Step	Hyp	Ref	Expression
1		ssv 3963	. 2 ⊢ 𝐴 ⊆ V
2		ssv 3963	. 2 ⊢ 𝐵 ⊆ V
3		xpss12 5667	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
4	1, 2, 3	mp2an 704	1 ⊢ (𝐴 × 𝐵) ⊆ (V × V)

Syntax hints:  Vcvv 3457   ⊆ wss 3907   × cxp 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3924  df-opab 5168  df-xp 5658

This theorem is referenced by:  relxp  5670  copsex2ga  5785  eqbrrdva  5846  relrelss  6264  dff3  7085  eqopi  8010  op1steq  8018  dfoprab4  8040  infxpenlem  9985  nqerf  10903  uzrdgfni  13985  reltrclfv  15044  homarel  18083  relxpchom  18227  frmdplusg  18903  psdmul  22289  upxp  23741  ustrel  24330  utop2nei  24368  utop3cls  24369  fmucndlem  24408  metustrel  24670  xppreima2  32908  df1stres  32961  df2ndres  32962  f1od2  32976  fsuppcurry1  32981  fsuppcurry2  32982  fpwrelmap  32990  metideq  34200  metider  34201  pstmfval  34203  xpinpreima2  34214  tpr2rico  34219  esum2d  34400  dya2iocnrect  34588  mpstssv  35902  txprel  36240  elxp8  37877  mblfinlem1  38168  xrnrel  38893  dihvalrel  41915  rfovcnvf1od  44592  ovolval2lem  47215  sprsymrelfo  48101