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Theorem pwtrrOLD 28917
Description: A set is transitive if its power set is. The proof of this theorem was automatically generated from pwtrrVD 28916 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Moved into main set.mm as pwtr 4242 and may be deleted by mathbox owner, AS. --NM 15-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
pwtrr.1  |-  A  e. 
_V
Assertion
Ref Expression
pwtrrOLD  |-  ( Tr 
~P A  ->  Tr  A )

Proof of Theorem pwtrrOLD
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . 6  |-  ( Tr 
~P A  ->  Tr  ~P A )
2 simpr 447 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  A )  ->  y  e.  A )
32a1i 10 . . . . . 6  |-  ( Tr 
~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  y  e.  A ) )
4 pwtrr.1 . . . . . . 7  |-  A  e. 
_V
54pwid 3651 . . . . . 6  |-  A  e. 
~P A
6 trel 4136 . . . . . . 7  |-  ( Tr 
~P A  ->  (
( y  e.  A  /\  A  e.  ~P A )  ->  y  e.  ~P A ) )
76exp3a 425 . . . . . 6  |-  ( Tr 
~P A  ->  (
y  e.  A  -> 
( A  e.  ~P A  ->  y  e.  ~P A ) ) )
81, 3, 5, 7ee120 28741 . . . . 5  |-  ( Tr 
~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  y  e.  ~P A ) )
9 elpwi 3646 . . . . 5  |-  ( y  e.  ~P A  -> 
y  C_  A )
108, 9syl6 29 . . . 4  |-  ( Tr 
~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  y  C_  A ) )
11 simpl 443 . . . . 5  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
1211a1i 10 . . . 4  |-  ( Tr 
~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  y ) )
13 ssel 3187 . . . 4  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
1410, 12, 13ee22 1352 . . 3  |-  ( Tr 
~P A  ->  (
( z  e.  y  /\  y  e.  A
)  ->  z  e.  A ) )
1514alrimivv 1622 . 2  |-  ( Tr 
~P A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
16 dftr2 4131 . 2  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
1715, 16sylibr 203 1  |-  ( Tr 
~P A  ->  Tr  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530    e. wcel 1696   _Vcvv 2801    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-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844  df-tr 4130
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