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Theorem trsspwALT 28993
Description: Virtual deduction proof of the left-to-right implication of dftr4 4309. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4309 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 3339 . . 3  |-  ( A 
C_  ~P A  <->  A. x
( x  e.  A  ->  x  e.  ~P A
) )
2 idn1 28727 . . . . . . 7  |-  (. Tr  A 
->.  Tr  A ).
3 idn2 28776 . . . . . . 7  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  A ).
4 trss 4313 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
52, 3, 4e12 28898 . . . . . 6  |-  (. Tr  A ,. x  e.  A  ->.  x 
C_  A ).
6 vex 2961 . . . . . . 7  |-  x  e. 
_V
76elpw 3807 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
85, 7e2bir 28796 . . . . 5  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  ~P A ).
98in2 28768 . . . 4  |-  (. Tr  A 
->.  ( x  e.  A  ->  x  e.  ~P A
) ).
109gen11 28779 . . 3  |-  (. Tr  A 
->.  A. x ( x  e.  A  ->  x  e.  ~P A ) ).
11 bi2 191 . . 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 28854 . 2  |-  (. Tr  A 
->.  A  C_  ~P A ).
1312in1 28724 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550    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  df-vd1 28723  df-vd2 28732
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