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Theorem xpss 4717
Description: A cross 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 3169 . 2 𝐴 ⊆ V
2 ssv 3169 . 2 𝐵 ⊆ V
3 xpss12 4716 . 2 ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
41, 2, 3mp2an 424 1 (𝐴 × 𝐵) ⊆ (V × V)
Colors of variables: wff set class
Syntax hints:  Vcvv 2730  wss 3121   × cxp 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-opab 4049  df-xp 4615
This theorem is referenced by:  relxp  4718  eqbrrdva  4779  relrelss  5135  funinsn  5245  eqopi  6148  op1steq  6155  dfoprab4  6168  f1od2  6211  frecuzrdgtcl  10355  frecuzrdgfunlem  10362  upxp  13025
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