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Theorem trsspwALT 31557
Description: Virtual deduction proof of the left-to-right implication of dftr4 4395. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4395 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 3350 . . 3  |-  ( A 
C_  ~P A  <->  A. x
( x  e.  A  ->  x  e.  ~P A
) )
2 idn1 31292 . . . . . . 7  |-  (. Tr  A 
->.  Tr  A ).
3 idn2 31340 . . . . . . 7  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  A ).
4 trss 4399 . . . . . . 7  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
52, 3, 4e12 31462 . . . . . 6  |-  (. Tr  A ,. x  e.  A  ->.  x 
C_  A ).
6 vex 2980 . . . . . . 7  |-  x  e. 
_V
76elpw 3871 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
85, 7e2bir 31360 . . . . 5  |-  (. Tr  A ,. x  e.  A  ->.  x  e.  ~P A ).
98in2 31332 . . . 4  |-  (. Tr  A 
->.  ( x  e.  A  ->  x  e.  ~P A
) ).
109gen11 31343 . . 3  |-  (. Tr  A 
->.  A. x ( x  e.  A  ->  x  e.  ~P A ) ).
11 bi2 198 . . 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 31418 . 2  |-  (. Tr  A 
->.  A  C_  ~P A ).
1312in1 31289 1  |-  ( Tr  A  ->  A  C_  ~P A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1367    e. wcel 1756    C_ wss 3333   ~Pcpw 3865   Tr wtr 4390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-v 2979  df-in 3340  df-ss 3347  df-pw 3867  df-uni 4097  df-tr 4391  df-vd1 31288  df-vd2 31296
This theorem is referenced by: (None)
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