Theorem xpss 5640

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 3947	. 2 ⊢ 𝐴 ⊆ V
2		ssv 3947	. 2 ⊢ 𝐵 ⊆ V
3		xpss12 5639	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
4	1, 2, 3	mp2an 693	1 ⊢ (𝐴 × 𝐵) ⊆ (V × V)

Syntax hints:  Vcvv 3430   ⊆ wss 3890   × cxp 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-ss 3907  df-opab 5149  df-xp 5630

This theorem is referenced by:  relxp  5642  copsex2ga  5756  eqbrrdva  5818  relrelss  6231  dff3  7046  eqopi  7971  op1steq  7979  dfoprab4  8001  infxpenlem  9926  nqerf  10844  uzrdgfni  13911  reltrclfv  14970  homarel  17994  relxpchom  18138  frmdplusg  18813  psdmul  22142  upxp  23598  ustrel  24187  utop2nei  24225  utop3cls  24226  fmucndlem  24265  metustrel  24527  xppreima2  32739  df1stres  32792  df2ndres  32793  f1od2  32807  fsuppcurry1  32812  fsuppcurry2  32813  fpwrelmap  32821  metideq  34053  metider  34054  pstmfval  34056  xpinpreima2  34067  tpr2rico  34072  esum2d  34253  dya2iocnrect  34441  mpstssv  35737  txprel  36075  elxp8  37701  mblfinlem1  37992  xrnrel  38717  dihvalrel  41739  rfovcnvf1od  44449  ovolval2lem  47089  sprsymrelfo  47969