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Theorem xpss12 4855
 Description: Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by NM, 26-Aug-1995.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
xpss12 ((A B C D) → (A × C) (B × D))

Proof of Theorem xpss12
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . . 4 (A B → (x Ax B))
2 ssel 3267 . . . 4 (C D → (y Cy D))
31, 2im2anan9 808 . . 3 ((A B C D) → ((x A y C) → (x B y D)))
43ssopab2dv 4715 . 2 ((A B C D) → {x, y (x A y C)} {x, y (x B y D)})
5 df-xp 4784 . 2 (A × C) = {x, y (x A y C)}
6 df-xp 4784 . 2 (B × D) = {x, y (x B y D)}
74, 5, 63sstr4g 3312 1 ((A B C D) → (A × C) (B × D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710   ⊆ wss 3257  {copab 4622   × cxp 4770 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-xp 4784 This theorem is referenced by:  xpss1  4856  xpss2  4857  ssxpb  5055  fssxp  5232
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