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Theorem pwsnALT 4011
Description: The power set of a singleton (direct proof). TO DO - should we keep this? (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pwsnALT  |-  ~P { A }  =  { (/)
,  { A } }

Proof of Theorem pwsnALT
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfss2 3338 . . . . . . . . 9  |-  ( x 
C_  { A }  <->  A. y ( y  e.  x  ->  y  e.  { A } ) )
2 elsn 3830 . . . . . . . . . . 11  |-  ( y  e.  { A }  <->  y  =  A )
32imbi2i 305 . . . . . . . . . 10  |-  ( ( y  e.  x  -> 
y  e.  { A } )  <->  ( y  e.  x  ->  y  =  A ) )
43albii 1576 . . . . . . . . 9  |-  ( A. y ( y  e.  x  ->  y  e.  { A } )  <->  A. y
( y  e.  x  ->  y  =  A ) )
51, 4bitri 242 . . . . . . . 8  |-  ( x 
C_  { A }  <->  A. y ( y  e.  x  ->  y  =  A ) )
6 neq0 3639 . . . . . . . . . 10  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
7 exintr 1625 . . . . . . . . . 10  |-  ( A. y ( y  e.  x  ->  y  =  A )  ->  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  y  =  A ) ) )
86, 7syl5bi 210 . . . . . . . . 9  |-  ( A. y ( y  e.  x  ->  y  =  A )  ->  ( -.  x  =  (/)  ->  E. y
( y  e.  x  /\  y  =  A
) ) )
9 df-clel 2433 . . . . . . . . . . 11  |-  ( A  e.  x  <->  E. y
( y  =  A  /\  y  e.  x
) )
10 exancom 1597 . . . . . . . . . . 11  |-  ( E. y ( y  =  A  /\  y  e.  x )  <->  E. y
( y  e.  x  /\  y  =  A
) )
119, 10bitr2i 243 . . . . . . . . . 10  |-  ( E. y ( y  e.  x  /\  y  =  A )  <->  A  e.  x )
12 snssi 3943 . . . . . . . . . 10  |-  ( A  e.  x  ->  { A }  C_  x )
1311, 12sylbi 189 . . . . . . . . 9  |-  ( E. y ( y  e.  x  /\  y  =  A )  ->  { A }  C_  x )
148, 13syl6 32 . . . . . . . 8  |-  ( A. y ( y  e.  x  ->  y  =  A )  ->  ( -.  x  =  (/)  ->  { A }  C_  x ) )
155, 14sylbi 189 . . . . . . 7  |-  ( x 
C_  { A }  ->  ( -.  x  =  (/)  ->  { A }  C_  x ) )
1615anc2li 542 . . . . . 6  |-  ( x 
C_  { A }  ->  ( -.  x  =  (/)  ->  ( x  C_  { A }  /\  { A }  C_  x ) ) )
17 eqss 3364 . . . . . 6  |-  ( x  =  { A }  <->  ( x  C_  { A }  /\  { A }  C_  x ) )
1816, 17syl6ibr 220 . . . . 5  |-  ( x 
C_  { A }  ->  ( -.  x  =  (/)  ->  x  =  { A } ) )
1918orrd 369 . . . 4  |-  ( x 
C_  { A }  ->  ( x  =  (/)  \/  x  =  { A } ) )
20 0ss 3657 . . . . . 6  |-  (/)  C_  { A }
21 sseq1 3370 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  { A }  <->  (/)  C_ 
{ A } ) )
2220, 21mpbiri 226 . . . . 5  |-  ( x  =  (/)  ->  x  C_  { A } )
23 eqimss 3401 . . . . 5  |-  ( x  =  { A }  ->  x  C_  { A } )
2422, 23jaoi 370 . . . 4  |-  ( ( x  =  (/)  \/  x  =  { A } )  ->  x  C_  { A } )
2519, 24impbii 182 . . 3  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
2625abbii 2549 . 2  |-  { x  |  x  C_  { A } }  =  {
x  |  ( x  =  (/)  \/  x  =  { A } ) }
27 df-pw 3802 . 2  |-  ~P { A }  =  {
x  |  x  C_  { A } }
28 dfpr2 3831 . 2  |-  { (/) ,  { A } }  =  { x  |  ( x  =  (/)  \/  x  =  { A } ) }
2926, 27, 283eqtr4i 2467 1  |-  ~P { A }  =  { (/)
,  { A } }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2423    C_ wss 3321   (/)c0 3629   ~Pcpw 3800   {csn 3815   {cpr 3816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-pw 3802  df-sn 3821  df-pr 3822
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