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Theorem pwsnALT 4246
Description: Alternate proof of pwsn 4245, 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 3488 . . . . . . . . 9  |-  ( x 
C_  { A }  <->  A. y ( y  e.  x  ->  y  e.  { A } ) )
2 elsn 4046 . . . . . . . . . . 11  |-  ( y  e.  { A }  <->  y  =  A )
32imbi2i 312 . . . . . . . . . 10  |-  ( ( y  e.  x  -> 
y  e.  { A } )  <->  ( y  e.  x  ->  y  =  A ) )
43albii 1641 . . . . . . . . 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 3804 . . . . . . . . . 10  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
7 exintr 1703 . . . . . . . . . 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 2452 . . . . . . . . . . 11  |-  ( A  e.  x  <->  E. y
( y  =  A  /\  y  e.  x
) )
10 exancom 1672 . . . . . . . . . . 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 4176 . . . . . . . . . 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 3514 . . . . . 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 3823 . . . . . 6  |-  (/)  C_  { A }
21 sseq1 3520 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  { A }  <->  (/)  C_ 
{ A } ) )
2220, 21mpbiri 233 . . . . 5  |-  ( x  =  (/)  ->  x  C_  { A } )
23 eqimss 3551 . . . . 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 2591 . 2  |-  { x  |  x  C_  { A } }  =  {
x  |  ( x  =  (/)  \/  x  =  { A } ) }
27 df-pw 4017 . 2  |-  ~P { A }  =  {
x  |  x  C_  { A } }
28 dfpr2 4047 . 2  |-  { (/) ,  { A } }  =  { x  |  ( x  =  (/)  \/  x  =  { A } ) }
2926, 27, 283eqtr4i 2496 1  |-  ~P { A }  =  { (/)
,  { A } }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369   A.wal 1393    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   {cpr 4034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-pw 4017  df-sn 4033  df-pr 4035
This theorem is referenced by: (None)
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