Theorem xpss 4719

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 3169	. 2 ⊢ 𝐴 ⊆ V
2		ssv 3169	. 2 ⊢ 𝐵 ⊆ V
3		xpss12 4718	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
4	1, 2, 3	mp2an 424	1 ⊢ (𝐴 × 𝐵) ⊆ (V × V)

Syntax hints:  Vcvv 2730   ⊆ wss 3121   × cxp 4609

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-opab 4051  df-xp 4617

This theorem is referenced by:  relxp  4720  eqbrrdva  4781  relrelss  5137  funinsn  5247  eqopi  6151  op1steq  6158  dfoprab4  6171  f1od2  6214  frecuzrdgtcl  10368  frecuzrdgfunlem  10375  upxp  13066