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Theorem trsspwALT 27860
Description: Virtual deduction proof of the left-to-right implication of dftr4 4119. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4119 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 )
Dummy variable  x is distinct from all other variables.

Proof of Theorem trsspwALT
StepHypRef Expression
1 dfss2 3170 . . 3  |-  ( A 
C_  ~P A  <->  A. x
( x  e.  A  ->  x  e.  ~P A
) )
2 idn1 27613 . . . . . . 7  |-  (. Tr  A 
->.  Tr  A ).
3 idn2 27653 . . . . . . 7  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  A ).
4 trss 4123 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
52, 3, 4e12 27767 . . . . . 6  |-  (. Tr  A ,. x  e.  A  ->.  x 
C_  A ).
6 vex 2792 . . . . . . 7  |-  x  e. 
_V
76elpw 3632 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
85, 7e2bir 27673 . . . . 5  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  ~P A ).
98in2 27645 . . . 4  |-  (. Tr  A 
->.  ( x  e.  A  ->  x  e.  ~P A
) ).
109gen11 27656 . . 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 27731 . 2  |-  (. Tr  A 
->.  A  C_  ~P A ).
1312in1 27610 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1528    e. wcel 1685    C_ wss 3153   ~Pcpw 3626   Tr wtr 4114
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-v 2791  df-in 3160  df-ss 3167  df-pw 3628  df-uni 3829  df-tr 4115  df-vd1 27609  df-vd2 27618
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