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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.
StepHypRef Expression
1 elxpi 5020 . . . 4
2 vex 3112 . . . . . . . 8
3 vex 3112 . . . . . . . 8
42, 3dfop 4216 . . . . . . 7
5 snssi 4174 . . . . . . . . . . . . 13
6 ssun3 3668 . . . . . . . . . . . . 13
75, 6syl 16 . . . . . . . . . . . 12
87adantr 465 . . . . . . . . . . 11
9 sseq1 3524 . . . . . . . . . . 11
108, 9syl5ibrcom 222 . . . . . . . . . 10
11 df-pr 4032 . . . . . . . . . . . 12
12 snssi 4174 . . . . . . . . . . . . . . 15
13 ssun4 3669 . . . . . . . . . . . . . . 15
1412, 13syl 16 . . . . . . . . . . . . . 14
157, 14anim12i 566 . . . . . . . . . . . . 13
16 unss 3677 . . . . . . . . . . . . 13
1715, 16sylib 196 . . . . . . . . . . . 12
1811, 17syl5eqss 3547 . . . . . . . . . . 11
19 sseq1 3524 . . . . . . . . . . 11
2018, 19syl5ibrcom 222 . . . . . . . . . 10
2110, 20jaod 380 . . . . . . . . 9
22 vex 3112 . . . . . . . . . 10
2322elpr 4047 . . . . . . . . 9
24 selpw 4019 . . . . . . . . 9
2521, 23, 243imtr4g 270 . . . . . . . 8
2625ssrdv 3509 . . . . . . 7
274, 26syl5eqss 3547 . . . . . 6
28 sseq1 3524 . . . . . . 7
2928biimpar 485 . . . . . 6
3027, 29sylan2 474 . . . . 5
3130exlimivv 1723 . . . 4
321, 31syl 16 . . 3
33 selpw 4019 . . 3
3432, 33sylibr 212 . 2
3534ssriv 3507 1
 Colors of variables: wff setvar class 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
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