Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwtrrVD Unicode version

Theorem pwtrrVD 28600
Description: Virtual deduction proof of pwtrrOLD 28601. (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 4115 . . 3  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
2 idn1 28342 . . . . . . . 8  |-  (. Tr  ~P A  ->.  Tr  ~P A ).
3 idn2 28385 . . . . . . . . 9  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  ( z  e.  y  /\  y  e.  A ) ).
4 simpr 447 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  A )  ->  y  e.  A )
53, 4e2 28403 . . . . . . . 8  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  e.  A ).
6 pwtrrVD.1 . . . . . . . . 9  |-  A  e. 
_V
76pwid 3638 . . . . . . . 8  |-  A  e. 
~P A
8 trel 4120 . . . . . . . . 9  |-  ( Tr 
~P A  ->  (
( y  e.  A  /\  A  e.  ~P A )  ->  y  e.  ~P A ) )
98exp3a 425 . . . . . . . 8  |-  ( Tr 
~P A  ->  (
y  e.  A  -> 
( A  e.  ~P A  ->  y  e.  ~P A ) ) )
102, 5, 7, 9e120 28435 . . . . . . 7  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  e.  ~P A ).
11 elpwi 3633 . . . . . . 7  |-  ( y  e.  ~P A  -> 
y  C_  A )
1210, 11e2 28403 . . . . . 6  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  C_  A ).
13 simpl 443 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
143, 13e2 28403 . . . . . 6  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  z  e.  y ).
15 ssel 3174 . . . . . 6  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
1612, 14, 15e22 28443 . . . . 5  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  z  e.  A ).
1716in2 28377 . . . 4  |-  (. Tr  ~P A  ->.  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
1817gen12 28390 . . 3  |-  (. Tr  ~P A  ->.  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
19 bi2 189 . . 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 28463 . 2  |-  (. Tr  ~P A  ->.  Tr  A ).
2120in1 28339 1  |-  ( Tr 
~P A  ->  Tr  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   Tr wtr 4113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828  df-tr 4114  df-vd1 28338  df-vd2 28347
  Copyright terms: Public domain W3C validator