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Theorem xpsspwOLD 5122
 Description: A Cartesian product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
xpsspwOLD

Proof of Theorem xpsspwOLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 5115 . 2
2 opelxp 5034 . . 3
3 snssi 4174 . . . . . . . 8
4 ssun3 3668 . . . . . . . 8
53, 4syl 16 . . . . . . 7
6 snex 4693 . . . . . . . 8
76elpw 4018 . . . . . . 7
85, 7sylibr 212 . . . . . 6
98adantr 465 . . . . 5
10 df-pr 4032 . . . . . . 7
11 snssi 4174 . . . . . . . . . 10
12 ssun4 3669 . . . . . . . . . 10
1311, 12syl 16 . . . . . . . . 9
145, 13anim12i 566 . . . . . . . 8
15 unss 3677 . . . . . . . 8
1614, 15sylib 196 . . . . . . 7
1710, 16syl5eqss 3547 . . . . . 6
18 zfpair2 4692 . . . . . . 7
1918elpw 4018 . . . . . 6
2017, 19sylibr 212 . . . . 5
219, 20jca 532 . . . 4
22 prex 4694 . . . . . 6
2322elpw 4018 . . . . 5
24 vex 3112 . . . . . . 7
25 vex 3112 . . . . . . 7
2624, 25dfop 4216 . . . . . 6
2726eleq1i 2534 . . . . 5
286, 18prss 4184 . . . . 5
2923, 27, 283bitr4ri 278 . . . 4
3021, 29sylib 196 . . 3
312, 30sylbi 195 . 2
321, 31relssi 5099 1
 Colors of variables: wff setvar class Syntax hints:  /\wa 369  e.wcel 1818  u.cun 3473  C_wss 3475  ~Pcpw 4012  {csn 4029  {cpr 4031  <.cop 4035  X.cxp 5002 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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 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-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  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  df-rel 5011
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