Theorem xpss 4655

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

Step	Hyp	Ref	Expression
1		ssv 3124	. 2 ⊢ 𝐴 ⊆ V
2		ssv 3124	. 2 ⊢ 𝐵 ⊆ V
3		xpss12 4654	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
4	1, 2, 3	mp2an 423	1 ⊢ (𝐴 × 𝐵) ⊆ (V × V)

Syntax hints:  Vcvv 2689   ⊆ wss 3076   × cxp 4545

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-opab 3998  df-xp 4553

This theorem is referenced by:  relxp  4656  eqbrrdva  4717  relrelss  5073  funinsn  5180  eqopi  6078  op1steq  6085  dfoprab4  6098  f1od2  6140  frecuzrdgtcl  10216  frecuzrdgfunlem  10223  upxp  12480