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Theorem trsspwALT 28908
Description: Virtual deduction proof of the left-to-right implication of dftr4 4134. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4134 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 3182 . . 3  |-  ( A 
C_  ~P A  <->  A. x
( x  e.  A  ->  x  e.  ~P A
) )
2 idn1 28641 . . . . . . 7  |-  (. Tr  A 
->.  Tr  A ).
3 idn2 28690 . . . . . . 7  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  A ).
4 trss 4138 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
52, 3, 4e12 28813 . . . . . 6  |-  (. Tr  A ,. x  e.  A  ->.  x 
C_  A ).
6 vex 2804 . . . . . . 7  |-  x  e. 
_V
76elpw 3644 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
85, 7e2bir 28710 . . . . 5  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  ~P A ).
98in2 28682 . . . 4  |-  (. Tr  A 
->.  ( x  e.  A  ->  x  e.  ~P A
) ).
109gen11 28693 . . 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 28768 . 2  |-  (. Tr  A 
->.  A  C_  ~P A ).
1312in1 28638 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    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  df-vd1 28637  df-vd2 28646
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