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Theorem trsspwALT 28592
Description: Virtual deduction proof of the left-to-right implication of dftr4 4118. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4118 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trsspwALT  |-  ( Tr  A  ->  A  C_  ~P A )

Proof of Theorem trsspwALT
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dfss2 3169 . . 3  |-  ( A 
C_  ~P A  <->  A. x
( x  e.  A  ->  x  e.  ~P A
) )
2 idn1 28342 . . . . . . 7  |-  (. Tr  A 
->.  Tr  A ).
3 idn2 28385 . . . . . . 7  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  A ).
4 trss 4122 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
52, 3, 4e12 28499 . . . . . 6  |-  (. Tr  A ,. x  e.  A  ->.  x 
C_  A ).
6 vex 2791 . . . . . . 7  |-  x  e. 
_V
76elpw 3631 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
85, 7e2bir 28405 . . . . 5  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  ~P A ).
98in2 28377 . . . 4  |-  (. Tr  A 
->.  ( x  e.  A  ->  x  e.  ~P A
) ).
109gen11 28388 . . 3  |-  (. Tr  A 
->.  A. x ( x  e.  A  ->  x  e.  ~P A ) ).
11 bi2 189 . . 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, 11e01 28463 . 2  |-  (. Tr  A 
->.  A  C_  ~P A ).
1312in1 28339 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   Tr wtr 4113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828  df-tr 4114  df-vd1 28338  df-vd2 28347
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