Theorem xpss 4771

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 3205	. 2 ⊢ 𝐴 ⊆ V
2		ssv 3205	. 2 ⊢ 𝐵 ⊆ V
3		xpss12 4770	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (V × V))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 × 𝐵) ⊆ (V × V)

Syntax hints:  Vcvv 2763   ⊆ wss 3157   × cxp 4661

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-opab 4095  df-xp 4669

This theorem is referenced by:  relxp  4772  eqbrrdva  4836  relrelss  5196  funinsn  5307  eqopi  6230  op1steq  6237  dfoprab4  6250  f1od2  6293  frecuzrdgtcl  10504  frecuzrdgfunlem  10511  reldvdsrsrg  13648  upxp  14508