Theorem xpss 4748

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 3191	. 2 ⊢ 𝐴 ⊆ V
2		ssv 3191	. 2 ⊢ 𝐵 ⊆ V
3		xpss12 4747	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 × 𝐵) ⊆ (V × V)

Syntax hints:  Vcvv 2751   ⊆ wss 3143   × cxp 4638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-in 3149  df-ss 3156  df-opab 4079  df-xp 4646

This theorem is referenced by:  relxp  4749  eqbrrdva  4811  relrelss  5169  funinsn  5279  eqopi  6190  op1steq  6197  dfoprab4  6210  f1od2  6253  frecuzrdgtcl  10429  frecuzrdgfunlem  10436  reldvdsrsrg  13402  upxp  14155