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Theorem xpss 4446
 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 2965 . 2 𝐴 ⊆ V
2 ssv 2965 . 2 𝐵 ⊆ V
3 xpss12 4445 . 2 ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
41, 2, 3mp2an 402 1 (𝐴 × 𝐵) ⊆ (V × V)
 Colors of variables: wff set class Syntax hints:  Vcvv 2557   ⊆ wss 2917   × cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-opab 3819  df-xp 4351 This theorem is referenced by:  relxp  4447  eqbrrdva  4505  relrelss  4844  eqopi  5798  op1steq  5805  dfoprab4  5818  frecuzrdgfn  9198
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