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Theorem pwtrrVD 37260
Description: Virtual deduction proof of pwtr 4666; see pwtrVD 37259 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
pwtrrVD.1  |-  A  e. 
_V
Assertion
Ref Expression
pwtrrVD  |-  ( Tr 
~P A  ->  Tr  A )

Proof of Theorem pwtrrVD
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 4512 . . 3  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
2 idn1 36986 . . . . . . . 8  |-  (. Tr  ~P A  ->.  Tr  ~P A ).
3 idn2 37034 . . . . . . . . 9  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  ( z  e.  y  /\  y  e.  A ) ).
4 simpr 467 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  A )  ->  y  e.  A )
53, 4e2 37052 . . . . . . . 8  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  e.  A ).
6 pwtrrVD.1 . . . . . . . . 9  |-  A  e. 
_V
76pwid 3976 . . . . . . . 8  |-  A  e. 
~P A
8 trel 4517 . . . . . . . . 9  |-  ( Tr 
~P A  ->  (
( y  e.  A  /\  A  e.  ~P A )  ->  y  e.  ~P A ) )
98expd 442 . . . . . . . 8  |-  ( Tr 
~P A  ->  (
y  e.  A  -> 
( A  e.  ~P A  ->  y  e.  ~P A ) ) )
102, 5, 7, 9e120 37084 . . . . . . 7  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  e.  ~P A ).
11 elpwi 3971 . . . . . . 7  |-  ( y  e.  ~P A  -> 
y  C_  A )
1210, 11e2 37052 . . . . . 6  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  C_  A ).
13 simpl 463 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
143, 13e2 37052 . . . . . 6  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  z  e.  y ).
15 ssel 3437 . . . . . 6  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
1612, 14, 15e22 37092 . . . . 5  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  z  e.  A ).
1716in2 37026 . . . 4  |-  (. Tr  ~P A  ->.  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
1817gen12 37039 . . 3  |-  (. Tr  ~P A  ->.  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
19 biimpr 203 . . 3  |-  ( ( Tr  A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )  -> 
( A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A )  ->  Tr  A ) )
201, 18, 19e01 37112 . 2  |-  (. Tr  ~P A  ->.  Tr  A ).
2120in1 36983 1  |-  ( Tr 
~P A  ->  Tr  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452    e. wcel 1897   _Vcvv 3056    C_ wss 3415   ~Pcpw 3962   Tr wtr 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-in 3422  df-ss 3429  df-pw 3964  df-uni 4212  df-tr 4511  df-vd1 36982  df-vd2 36990
This theorem is referenced by: (None)
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