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Theorem pwnss 4617
Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
pwnss

Proof of Theorem pwnss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq12 2533 . . . . . . 7
21anidms 645 . . . . . 6
32notbid 294 . . . . 5
4 df-nel 2655 . . . . . . 7
5 eleq12 2533 . . . . . . . . 9
65anidms 645 . . . . . . . 8
76notbid 294 . . . . . . 7
84, 7syl5bb 257 . . . . . 6
98cbvrabv 3108 . . . . 5
103, 9elrab2 3259 . . . 4
11 pclem6 930 . . . 4
1210, 11ax-mp 5 . . 3
13 ssel 3497 . . 3
1412, 13mtoi 178 . 2
15 ssrab2 3584 . . 3
16 elpw2g 4615 . . 3
1715, 16mpbiri 233 . 2
1814, 17nsyl3 119 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  e/wnel 2653  {crab 2811  C_wss 3475  ~Pcpw 4012
This theorem is referenced by:  pwne  4618  pwuninel2  7022
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  ax-sep 4573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-nel 2655  df-rab 2816  df-v 3111  df-in 3482  df-ss 3489  df-pw 4014
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