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Theorem xpss 4474
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 2993 . 2 𝐴 ⊆ V
2 ssv 2993 . 2 𝐵 ⊆ V
3 xpss12 4473 . 2 ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
41, 2, 3mp2an 410 1 (𝐴 × 𝐵) ⊆ (V × V)
Colors of variables: wff set class
Syntax hints:  Vcvv 2574  wss 2945   × cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-opab 3847  df-xp 4379
This theorem is referenced by:  relxp  4475  eqbrrdva  4533  relrelss  4872  eqopi  5826  op1steq  5833  dfoprab4  5846  f1od2  5884  frecuzrdgfn  9362
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