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Theorem trsspwALT2 32990
Description: Virtual deduction proof of trsspwALT 32989. This proof is the same as the proof of trsspwALT 32989 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 3498 . . 3  |-  ( A 
C_  ~P A  <->  A. x
( x  e.  A  ->  x  e.  ~P A
) )
2 id 22 . . . . . . 7  |-  ( Tr  A  ->  Tr  A
)
3 idd 24 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  e.  A ) )
4 trss 4554 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
52, 3, 4sylsyld 56 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
6 vex 3121 . . . . . . 7  |-  x  e. 
_V
76elpw 4021 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
85, 7syl6ibr 227 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  x  e. 
~P A ) )
98idiALT 32590 . . . 4  |-  ( Tr  A  ->  ( x  e.  A  ->  x  e. 
~P A ) )
109alrimiv 1695 . . 3  |-  ( Tr  A  ->  A. x
( x  e.  A  ->  x  e.  ~P A
) )
11 bi2 198 . . 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 63 . 2  |-  ( Tr  A  ->  A  C_  ~P A )
1312idiALT 32590 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    e. wcel 1767    C_ wss 3481   ~Pcpw 4015   Tr wtr 4545
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-an 371  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-ral 2822  df-v 3120  df-in 3488  df-ss 3495  df-pw 4017  df-uni 4251  df-tr 4546
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator