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Theorem pwtrVD 37260
Description: Virtual deduction proof of pwtr 4667; see pwtrrVD 37261 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pwtrVD  |-  ( Tr  A  ->  Tr  ~P A
)

Proof of Theorem pwtrVD
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 4513 . . 3  |-  ( Tr 
~P A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  ~P A )  -> 
z  e.  ~P A
) )
2 idn1 36987 . . . . . . 7  |-  (. Tr  A 
->.  Tr  A ).
3 idn2 37035 . . . . . . . . . 10  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  ( z  e.  y  /\  y  e.  ~P A ) ).
4 simpr 467 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  ~P A
)  ->  y  e.  ~P A )
53, 4e2 37053 . . . . . . . . 9  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  y  e.  ~P A ).
6 elpwi 3972 . . . . . . . . 9  |-  ( y  e.  ~P A  -> 
y  C_  A )
75, 6e2 37053 . . . . . . . 8  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  y  C_  A ).
8 simpl 463 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  ~P A
)  ->  z  e.  y )
93, 8e2 37053 . . . . . . . 8  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  y ).
10 ssel 3438 . . . . . . . 8  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
117, 9, 10e22 37093 . . . . . . 7  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  A ).
12 trss 4520 . . . . . . 7  |-  ( Tr  A  ->  ( z  e.  A  ->  z  C_  A ) )
132, 11, 12e12 37151 . . . . . 6  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  C_  A ).
14 vex 3060 . . . . . . 7  |-  z  e. 
_V
1514elpw 3969 . . . . . 6  |-  ( z  e.  ~P A  <->  z  C_  A )
1613, 15e2bir 37055 . . . . 5  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  ~P A ).
1716in2 37027 . . . 4  |-  (. Tr  A 
->.  ( ( z  e.  y  /\  y  e. 
~P A )  -> 
z  e.  ~P A
) ).
1817gen12 37040 . . 3  |-  (. Tr  A 
->.  A. z A. y
( ( z  e.  y  /\  y  e. 
~P A )  -> 
z  e.  ~P A
) ).
19 biimpr 203 . . 3  |-  ( ( Tr  ~P A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  ~P A )  -> 
z  e.  ~P A
) )  ->  ( A. z A. y ( ( z  e.  y  /\  y  e.  ~P A )  ->  z  e.  ~P A )  ->  Tr  ~P A ) )
201, 18, 19e01 37113 . 2  |-  (. Tr  A 
->.  Tr  ~P A ).
2120in1 36984 1  |-  ( Tr  A  ->  Tr  ~P A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1453    e. wcel 1898    C_ wss 3416   ~Pcpw 3963   Tr wtr 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-v 3059  df-in 3423  df-ss 3430  df-pw 3965  df-uni 4213  df-tr 4512  df-vd1 36983  df-vd2 36991
This theorem is referenced by: (None)
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