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Theorem trsspwALT2 37270
Description: Virtual deduction proof of trsspwALT 37269. This proof is the same as the proof of trsspwALT 37269 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 3407 . . 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 4499 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
52, 3, 4sylsyld 57 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
6 vex 3034 . . . . . . 7  |-  x  e. 
_V
76elpw 3948 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
85, 7syl6ibr 235 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  x  e. 
~P A ) )
98idiALT 36902 . . . 4  |-  ( Tr  A  ->  ( x  e.  A  ->  x  e. 
~P A ) )
109alrimiv 1781 . . 3  |-  ( Tr  A  ->  A. x
( x  e.  A  ->  x  e.  ~P A
) )
11 biimpr 203 . . 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 64 . 2  |-  ( Tr  A  ->  A  C_  ~P A )
1312idiALT 36902 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450    e. wcel 1904    C_ wss 3390   ~Pcpw 3942   Tr wtr 4490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-v 3033  df-in 3397  df-ss 3404  df-pw 3944  df-uni 4191  df-tr 4491
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator