Theorem xpss 4575

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 3061	. 2 ⊢ 𝐴 ⊆ V
2		ssv 3061	. 2 ⊢ 𝐵 ⊆ V
3		xpss12 4574	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
4	1, 2, 3	mp2an 418	1 ⊢ (𝐴 × 𝐵) ⊆ (V × V)

Syntax hints:  Vcvv 2633   ⊆ wss 3013   × cxp 4465

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-opab 3922  df-xp 4473

This theorem is referenced by:  relxp  4576  eqbrrdva  4637  relrelss  4991  funinsn  5097  eqopi  5980  op1steq  5987  dfoprab4  6000  f1od2  6038  frecuzrdgtcl  9968  frecuzrdgfunlem  9975