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Theorem xpss 5564
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 3988 . 2 𝐴 ⊆ V
2 ssv 3988 . 2 𝐵 ⊆ V
3 xpss12 5563 . 2 ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
41, 2, 3mp2an 688 1 (𝐴 × 𝐵) ⊆ (V × V)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3492  wss 3933   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-ss 3949  df-opab 5120  df-xp 5554
This theorem is referenced by:  relxp  5566  copsex2ga  5673  eqbrrdva  5733  relrelss  6117  dff3  6858  eqopi  7714  op1steq  7722  dfoprab4  7742  infxpenlem  9427  nqerf  10340  uzrdgfni  13314  reltrclfv  14365  homarel  17284  relxpchom  17419  frmdplusg  18007  upxp  22159  ustrel  22747  utop2nei  22786  utop3cls  22787  fmucndlem  22827  metustrel  23089  xppreima2  30323  df1stres  30365  df2ndres  30366  f1od2  30383  fsuppcurry1  30387  fsuppcurry2  30388  fpwrelmap  30395  metideq  31032  metider  31033  pstmfval  31035  xpinpreima2  31049  tpr2rico  31054  esum2d  31251  dya2iocnrect  31438  mpstssv  32683  txprel  33237  elxp8  34534  mblfinlem1  34810  xrnrel  35505  dihvalrel  38295  rfovcnvf1od  40228  ovolval2lem  42802  sprsymrelfo  43536
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