Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trsspwALT3 Unicode version

Theorem trsspwALT3 28910
Description: Short predicate calculus proof of the left-to-right implication of dftr4 4134. 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 28909, which is the virtual deduction proof trsspwALT 28908 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 4138 . . 3  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
2 vex 2804 . . . 4  |-  x  e. 
_V
32elpw 3644 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
41, 3syl6ibr 218 . 2  |-  ( Tr  A  ->  ( x  e.  A  ->  x  e. 
~P A ) )
54ssrdv 3198 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   Tr wtr 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844  df-tr 4130
  Copyright terms: Public domain W3C validator