Theorem xpss12 4856

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 C_ B /\ C C_ D) -> (A X. C) C_ (B X. D))

Proof of Theorem xpss12

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . 4 |- (A C_ B -> (x e. A -> x e. B))
2		ssel 3268	. . . 4 |- (C C_ D -> (y e. C -> y e. D))
3	1, 2	im2anan9 808	. . 3 |- ((A C_ B /\ C C_ D) -> ((x e. A /\ y e. C) -> (x e. B /\ y e. D)))
4	3	ssopab2dv 4716	. 2 |- ((A C_ B /\ C C_ D) -> {<.x, y>. | (x e. A /\ y e. C)} C_ {<.x, y>. | (x e. B /\ y e. D)})
5		df-xp 4785	. 2 |- (A X. C) = {<.x, y>. | (x e. A /\ y e. C)}
6		df-xp 4785	. 2 |- (B X. D) = {<.x, y>. | (x e. B /\ y e. D)}
7	4, 5, 6	3sstr4g 3313	1 |- ((A C_ B /\ C C_ D) -> (A X. C) C_ (B X. D))

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710   C_ wss 3258  {copab 4623   X. cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-xp 4785

This theorem is referenced by:  xpss1  4857  xpss2  4858  ssxpb  5056  fssxp  5233