Theorem xpss 5226

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 3625	. 2 ⊢ 𝐴 ⊆ V
2		ssv 3625	. 2 ⊢ 𝐵 ⊆ V
3		xpss12 5225	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
4	1, 2, 3	mp2an 708	1 ⊢ (𝐴 × 𝐵) ⊆ (V × V)

Syntax hints:  Vcvv 3200   ⊆ wss 3574   × cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-opab 4713  df-xp 5120

This theorem is referenced by:  relxp  5227  copsex2ga  5231  eqbrrdva  5291  relrelss  5659  dff3  6372  eqopi  7202  op1steq  7210  dfoprab4  7225  infxpenlem  8836  nqerf  9752  uzrdgfni  12757  reltrclfv  13758  homarel  16686  relxpchom  16821  frmdplusg  17391  upxp  21426  ustrel  22015  utop2nei  22054  utop3cls  22055  fmucndlem  22095  metustrel  22357  xppreima2  29450  df1stres  29481  df2ndres  29482  f1od2  29499  fpwrelmap  29508  metideq  29936  metider  29937  pstmfval  29939  xpinpreima2  29953  tpr2rico  29958  esum2d  30155  dya2iocnrect  30343  mpstssv  31436  txprel  31986  bj-elid2  33086  elxp8  33219  mblfinlem1  33446  xrnrel  34136  dihvalrel  36568  rfovcnvf1od  38298  ovolval2lem  40857  sprsymrelfo  41747