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Theorem xpss 4546
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 3046 . 2 𝐴 ⊆ V
2 ssv 3046 . 2 𝐵 ⊆ V
3 xpss12 4545 . 2 ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
41, 2, 3mp2an 417 1 (𝐴 × 𝐵) ⊆ (V × V)
Colors of variables: wff set class
Syntax hints:  Vcvv 2619  wss 2999   × cxp 4436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-opab 3900  df-xp 4444
This theorem is referenced by:  relxp  4547  eqbrrdva  4606  relrelss  4957  funinsn  5063  eqopi  5942  op1steq  5949  dfoprab4  5962  f1od2  6000  frecuzrdgtcl  9819  frecuzrdgfunlem  9826
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