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Theorem pwnss 4566
 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 2530 . . . . . . 7
21anidms 645 . . . . . 6
32notbid 294 . . . . 5
4 df-nel 2651 . . . . . . 7
5 eleq12 2530 . . . . . . . . 9
65anidms 645 . . . . . . . 8
76notbid 294 . . . . . . 7
84, 7syl5bb 257 . . . . . 6
98cbvrabv 3077 . . . . 5
103, 9elrab2 3226 . . . 4
11 pclem6 921 . . . 4
1210, 11ax-mp 5 . . 3
13 ssel 3459 . . 3
1412, 13mtoi 178 . 2
15 ssrab2 3546 . . 3
16 elpw2g 4564 . . 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 1370   wcel 1758   wnel 2649  crab 2803   wss 3437  cpw 3969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-nel 2651  df-rab 2808  df-v 3080  df-in 3444  df-ss 3451  df-pw 3971 This theorem is referenced by:  pwne  4567  pwuninel2  6904
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