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Theorem pwtrrVD 31861
Description: Virtual deduction proof of pwtr 4643; see pwtrVD 31860 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 4485 . . 3  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
2 idn1 31587 . . . . . . . 8  |-  (. Tr  ~P A  ->.  Tr  ~P A ).
3 idn2 31635 . . . . . . . . 9  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  ( z  e.  y  /\  y  e.  A ) ).
4 simpr 461 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  A )  ->  y  e.  A )
53, 4e2 31653 . . . . . . . 8  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  e.  A ).
6 pwtrrVD.1 . . . . . . . . 9  |-  A  e. 
_V
76pwid 3972 . . . . . . . 8  |-  A  e. 
~P A
8 trel 4490 . . . . . . . . 9  |-  ( Tr 
~P A  ->  (
( y  e.  A  /\  A  e.  ~P A )  ->  y  e.  ~P A ) )
98expd 436 . . . . . . . 8  |-  ( Tr 
~P A  ->  (
y  e.  A  -> 
( A  e.  ~P A  ->  y  e.  ~P A ) ) )
102, 5, 7, 9e120 31685 . . . . . . 7  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  e.  ~P A ).
11 elpwi 3967 . . . . . . 7  |-  ( y  e.  ~P A  -> 
y  C_  A )
1210, 11e2 31653 . . . . . 6  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  C_  A ).
13 simpl 457 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
143, 13e2 31653 . . . . . 6  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  z  e.  y ).
15 ssel 3448 . . . . . 6  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
1612, 14, 15e22 31693 . . . . 5  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  z  e.  A ).
1716in2 31627 . . . 4  |-  (. Tr  ~P A  ->.  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
1817gen12 31640 . . 3  |-  (. Tr  ~P A  ->.  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
19 bi2 198 . . 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 31713 . 2  |-  (. Tr  ~P A  ->.  Tr  A ).
2120in1 31584 1  |-  ( Tr 
~P A  ->  Tr  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    e. wcel 1758   _Vcvv 3068    C_ wss 3426   ~Pcpw 3958   Tr wtr 4483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3070  df-in 3433  df-ss 3440  df-pw 3960  df-uni 4190  df-tr 4484  df-vd1 31583  df-vd2 31591
This theorem is referenced by: (None)
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