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Theorem xpss 5295
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
StepHypRef Expression
1 ssv 3787 . 2 𝐴 ⊆ V
2 ssv 3787 . 2 𝐵 ⊆ V
3 xpss12 5294 . 2 ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
41, 2, 3mp2an 683 1 (𝐴 × 𝐵) ⊆ (V × V)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3350  wss 3734   × cxp 5277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-in 3741  df-ss 3748  df-opab 4874  df-xp 5285
This theorem is referenced by:  relxp  5297  copsex2ga  5401  eqbrrdva  5462  relrelss  5847  dff3  6566  eqopi  7406  op1steq  7414  dfoprab4  7429  infxpenlem  9091  nqerf  10009  uzrdgfni  12970  reltrclfv  14057  homarel  16965  relxpchom  17101  frmdplusg  17672  upxp  21720  ustrel  22308  utop2nei  22347  utop3cls  22348  fmucndlem  22388  metustrel  22650  xppreima2  29921  df1stres  29951  df2ndres  29952  f1od2  29969  fpwrelmap  29978  metideq  30404  metider  30405  pstmfval  30407  xpinpreima2  30421  tpr2rico  30426  esum2d  30623  dya2iocnrect  30811  mpstssv  31905  txprel  32451  bj-elid2  33540  elxp8  33673  mblfinlem1  33891  xrnrel  34585  dihvalrel  37256  rfovcnvf1od  38996  ovolval2lem  41521  sprsymrelfo  42440
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