Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trsspwALT2 Unicode version

Theorem trsspwALT2 28273
Description: Virtual deduction proof of trsspwALT 28272. This proof is the same as the proof of trsspwALT 28272 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 3280 . . 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 4252 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
52, 3, 4sylsyld 54 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
6 vex 2902 . . . . . . 7  |-  x  e. 
_V
76elpw 3748 . . . . . 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 1638 . . 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 1546    e. wcel 1717    C_ wss 3263   ~Pcpw 3742   Tr wtr 4243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-v 2901  df-in 3270  df-ss 3277  df-pw 3744  df-uni 3958  df-tr 4244
  Copyright terms: Public domain W3C validator