Theorem xpss 4787

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 3216	. 2 ⊢ 𝐴 ⊆ V
2		ssv 3216	. 2 ⊢ 𝐵 ⊆ V
3		xpss12 4786	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 × 𝐵) ⊆ (V × V)

Syntax hints:  Vcvv 2773   ⊆ wss 3167   × cxp 4677

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3173  df-ss 3180  df-opab 4110  df-xp 4685

This theorem is referenced by:  relxp  4788  eqbrrdva  4852  relrelss  5214  funinsn  5328  eqopi  6265  op1steq  6272  dfoprab4  6285  f1od2  6328  frecuzrdgtcl  10564  frecuzrdgfunlem  10571  reldvdsrsrg  13898  upxp  14788