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Theorem xpsspw 5114
 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.
StepHypRef Expression
1 elxpi 5015 . . . 4
2 vex 3116 . . . . . . . 8
3 vex 3116 . . . . . . . 8
42, 3dfop 4212 . . . . . . 7
5 snssi 4171 . . . . . . . . . . . . 13
6 ssun3 3669 . . . . . . . . . . . . 13
75, 6syl 16 . . . . . . . . . . . 12
87adantr 465 . . . . . . . . . . 11
9 sseq1 3525 . . . . . . . . . . 11
108, 9syl5ibrcom 222 . . . . . . . . . 10
11 df-pr 4030 . . . . . . . . . . . 12
12 snssi 4171 . . . . . . . . . . . . . . 15
13 ssun4 3670 . . . . . . . . . . . . . . 15
1412, 13syl 16 . . . . . . . . . . . . . 14
157, 14anim12i 566 . . . . . . . . . . . . 13
16 unss 3678 . . . . . . . . . . . . 13
1715, 16sylib 196 . . . . . . . . . . . 12
1811, 17syl5eqss 3548 . . . . . . . . . . 11
19 sseq1 3525 . . . . . . . . . . 11
2018, 19syl5ibrcom 222 . . . . . . . . . 10
2110, 20jaod 380 . . . . . . . . 9
22 vex 3116 . . . . . . . . . 10
2322elpr 4045 . . . . . . . . 9
24 selpw 4017 . . . . . . . . 9
2521, 23, 243imtr4g 270 . . . . . . . 8
2625ssrdv 3510 . . . . . . 7
274, 26syl5eqss 3548 . . . . . 6
28 sseq1 3525 . . . . . . 7
2928biimpar 485 . . . . . 6
3027, 29sylan2 474 . . . . 5
3130exlimivv 1699 . . . 4
321, 31syl 16 . . 3
33 selpw 4017 . . 3
3432, 33sylibr 212 . 2
3534ssriv 3508 1
 Colors of variables: wff setvar class Syntax hints:   wo 368   wa 369   wceq 1379  wex 1596   wcel 1767   cun 3474   wss 3476  cpw 4010  csn 4027  cpr 4029  cop 4033   cxp 4997 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005 This theorem is referenced by:  unixpss  5116  xpexg  6709  rankxpu  8290  wunxp  9098  gruxp  9181
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