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Theorem trsspwALT3 28594
Description: Short predicate calculus proof of the left-to-right implication of dftr4 4118. A transitive class is a subset of its power class. This proof was constructed by applying Metamath's minimize command to the proof of trsspwALT2 28593, which is the virtual deduction proof trsspwALT 28592 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trsspwALT3  |-  ( Tr  A  ->  A  C_  ~P A )

Proof of Theorem trsspwALT3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 trss 4122 . . 3  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
2 vex 2791 . . . 4  |-  x  e. 
_V
32elpw 3631 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
41, 3syl6ibr 218 . 2  |-  ( Tr  A  ->  ( x  e.  A  ->  x  e. 
~P A ) )
54ssrdv 3185 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    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
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