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Theorem pwsnALT 4240
Description: Alternate proof of pwsn 4239, more direct. (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 3493 . . . . . . . . 9  |-  ( x 
C_  { A }  <->  A. y ( y  e.  x  ->  y  e.  { A } ) )
2 elsn 4041 . . . . . . . . . . 11  |-  ( y  e.  { A }  <->  y  =  A )
32imbi2i 312 . . . . . . . . . 10  |-  ( ( y  e.  x  -> 
y  e.  { A } )  <->  ( y  e.  x  ->  y  =  A ) )
43albii 1620 . . . . . . . . 9  |-  ( A. y ( y  e.  x  ->  y  e.  { A } )  <->  A. y
( y  e.  x  ->  y  =  A ) )
51, 4bitri 249 . . . . . . . 8  |-  ( x 
C_  { A }  <->  A. y ( y  e.  x  ->  y  =  A ) )
6 neq0 3795 . . . . . . . . . 10  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
7 exintr 1678 . . . . . . . . . 10  |-  ( A. y ( y  e.  x  ->  y  =  A )  ->  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  y  =  A ) ) )
86, 7syl5bi 217 . . . . . . . . 9  |-  ( A. y ( y  e.  x  ->  y  =  A )  ->  ( -.  x  =  (/)  ->  E. y
( y  e.  x  /\  y  =  A
) ) )
9 df-clel 2462 . . . . . . . . . . 11  |-  ( A  e.  x  <->  E. y
( y  =  A  /\  y  e.  x
) )
10 exancom 1648 . . . . . . . . . . 11  |-  ( E. y ( y  =  A  /\  y  e.  x )  <->  E. y
( y  e.  x  /\  y  =  A
) )
119, 10bitr2i 250 . . . . . . . . . 10  |-  ( E. y ( y  e.  x  /\  y  =  A )  <->  A  e.  x )
12 snssi 4171 . . . . . . . . . 10  |-  ( A  e.  x  ->  { A }  C_  x )
1311, 12sylbi 195 . . . . . . . . 9  |-  ( E. y ( y  e.  x  /\  y  =  A )  ->  { A }  C_  x )
148, 13syl6 33 . . . . . . . 8  |-  ( A. y ( y  e.  x  ->  y  =  A )  ->  ( -.  x  =  (/)  ->  { A }  C_  x ) )
155, 14sylbi 195 . . . . . . 7  |-  ( x 
C_  { A }  ->  ( -.  x  =  (/)  ->  { A }  C_  x ) )
1615anc2li 557 . . . . . 6  |-  ( x 
C_  { A }  ->  ( -.  x  =  (/)  ->  ( x  C_  { A }  /\  { A }  C_  x ) ) )
17 eqss 3519 . . . . . 6  |-  ( x  =  { A }  <->  ( x  C_  { A }  /\  { A }  C_  x ) )
1816, 17syl6ibr 227 . . . . 5  |-  ( x 
C_  { A }  ->  ( -.  x  =  (/)  ->  x  =  { A } ) )
1918orrd 378 . . . 4  |-  ( x 
C_  { A }  ->  ( x  =  (/)  \/  x  =  { A } ) )
20 0ss 3814 . . . . . 6  |-  (/)  C_  { A }
21 sseq1 3525 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  { A }  <->  (/)  C_ 
{ A } ) )
2220, 21mpbiri 233 . . . . 5  |-  ( x  =  (/)  ->  x  C_  { A } )
23 eqimss 3556 . . . . 5  |-  ( x  =  { A }  ->  x  C_  { A } )
2422, 23jaoi 379 . . . 4  |-  ( ( x  =  (/)  \/  x  =  { A } )  ->  x  C_  { A } )
2519, 24impbii 188 . . 3  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
2625abbii 2601 . 2  |-  { x  |  x  C_  { A } }  =  {
x  |  ( x  =  (/)  \/  x  =  { A } ) }
27 df-pw 4012 . 2  |-  ~P { A }  =  {
x  |  x  C_  { A } }
28 dfpr2 4042 . 2  |-  { (/) ,  { A } }  =  { x  |  ( x  =  (/)  \/  x  =  { A } ) }
2926, 27, 283eqtr4i 2506 1  |-  ~P { A }  =  { (/)
,  { A } }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   {cpr 4029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-pw 4012  df-sn 4028  df-pr 4030
This theorem is referenced by: (None)
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