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Theorem trsspwALT2 28833
Description: Virtual deduction proof of trsspwALT 28832. This proof is the same as the proof of trsspwALT 28832 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A transitive class is a subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trsspwALT2  |-  ( Tr  A  ->  A  C_  ~P A )

Proof of Theorem trsspwALT2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dfss2 3329 . . 3  |-  ( A 
C_  ~P A  <->  A. x
( x  e.  A  ->  x  e.  ~P A
) )
2 id 20 . . . . . . 7  |-  ( Tr  A  ->  Tr  A
)
3 idd 22 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  e.  A ) )
4 trss 4303 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
52, 3, 4sylsyld 54 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
6 vex 2951 . . . . . . 7  |-  x  e. 
_V
76elpw 3797 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
85, 7syl6ibr 219 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  x  e. 
~P A ) )
98idi 2 . . . 4  |-  ( Tr  A  ->  ( x  e.  A  ->  x  e. 
~P A ) )
109alrimiv 1641 . . 3  |-  ( Tr  A  ->  A. x
( x  e.  A  ->  x  e.  ~P A
) )
11 bi2 190 . . 3  |-  ( ( A  C_  ~P A  <->  A. x ( x  e.  A  ->  x  e.  ~P A ) )  -> 
( A. x ( x  e.  A  ->  x  e.  ~P A
)  ->  A  C_  ~P A ) )
121, 10, 11mpsyl 61 . 2  |-  ( Tr  A  ->  A  C_  ~P A )
1312idi 2 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    e. wcel 1725    C_ wss 3312   ~Pcpw 3791   Tr wtr 4294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793  df-uni 4008  df-tr 4295
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