Theorem xpsspw 5121

Description: A Cartesian product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)

Assertion

Ref	Expression
xpsspw

Proof of Theorem xpsspw

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

Step	Hyp	Ref	Expression
1		elxpi 5020	. . . 4
2		vex 3112	. . . . . . . 8
3		vex 3112	. . . . . . . 8
4	2, 3	dfop 4216	. . . . . . 7
5		snssi 4174	. . . . . . . . . . . . 13
6		ssun3 3668	. . . . . . . . . . . . 13
7	5, 6	syl 16	. . . . . . . . . . . 12
8	7	adantr 465	. . . . . . . . . . 11
9		sseq1 3524	. . . . . . . . . . 11
10	8, 9	syl5ibrcom 222	. . . . . . . . . 10
11		df-pr 4032	. . . . . . . . . . . 12
12		snssi 4174	. . . . . . . . . . . . . . 15
13		ssun4 3669	. . . . . . . . . . . . . . 15
14	12, 13	syl 16	. . . . . . . . . . . . . 14
15	7, 14	anim12i 566	. . . . . . . . . . . . 13
16		unss 3677	. . . . . . . . . . . . 13
17	15, 16	sylib 196	. . . . . . . . . . . 12
18	11, 17	syl5eqss 3547	. . . . . . . . . . 11
19		sseq1 3524	. . . . . . . . . . 11
20	18, 19	syl5ibrcom 222	. . . . . . . . . 10
21	10, 20	jaod 380	. . . . . . . . 9
22		vex 3112	. . . . . . . . . 10
23	22	elpr 4047	. . . . . . . . 9
24		selpw 4019	. . . . . . . . 9
25	21, 23, 24	3imtr4g 270	. . . . . . . 8
26	25	ssrdv 3509	. . . . . . 7
27	4, 26	syl5eqss 3547	. . . . . 6
28		sseq1 3524	. . . . . . 7
29	28	biimpar 485	. . . . . 6
30	27, 29	sylan2 474	. . . . 5
31	30	exlimivv 1723	. . . 4
32	1, 31	syl 16	. . 3
33		selpw 4019	. . 3
34	32, 33	sylibr 212	. 2
35	34	ssriv 3507	1

Syntax hints:  \/wo 368  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  u.cun 3473  C_wss 3475  ~Pcpw 4012  {csn 4029  {cpr 4031  <.cop 4035  X.cxp 5002

This theorem is referenced by:  unixpss  5123  xpexg  6602  rankxpu  8315  wunxp  9123  gruxp  9206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511  df-xp 5010