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Theorem trsspwALT 28640
Description: Virtual deduction proof of the left-to-right implication of dftr4 4267. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4267 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 3297 . . 3  |-  ( A 
C_  ~P A  <->  A. x
( x  e.  A  ->  x  e.  ~P A
) )
2 idn1 28374 . . . . . . 7  |-  (. Tr  A 
->.  Tr  A ).
3 idn2 28423 . . . . . . 7  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  A ).
4 trss 4271 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
52, 3, 4e12 28545 . . . . . 6  |-  (. Tr  A ,. x  e.  A  ->.  x 
C_  A ).
6 vex 2919 . . . . . . 7  |-  x  e. 
_V
76elpw 3765 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
85, 7e2bir 28443 . . . . 5  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  ~P A ).
98in2 28415 . . . 4  |-  (. Tr  A 
->.  ( x  e.  A  ->  x  e.  ~P A
) ).
109gen11 28426 . . 3  |-  (. Tr  A 
->.  A. x ( x  e.  A  ->  x  e.  ~P A ) ).
11 bi2 190 . . 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 28501 . 2  |-  (. Tr  A 
->.  A  C_  ~P A ).
1312in1 28371 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    e. wcel 1721    C_ wss 3280   ~Pcpw 3759   Tr wtr 4262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-in 3287  df-ss 3294  df-pw 3761  df-uni 3976  df-tr 4263  df-vd1 28370  df-vd2 28379
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