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Theorem trsspwALT3 28995
Description: Short predicate calculus proof of the left-to-right implication of dftr4 4309. 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 28994, which is the virtual deduction proof trsspwALT 28993 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 4313 . . 3  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
2 vex 2961 . . . 4  |-  x  e. 
_V
32elpw 3807 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
41, 3syl6ibr 220 . 2  |-  ( Tr  A  ->  ( x  e.  A  ->  x  e. 
~P A ) )
54ssrdv 3356 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726    C_ wss 3322   ~Pcpw 3801   Tr wtr 4304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-in 3329  df-ss 3336  df-pw 3803  df-uni 4018  df-tr 4305
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