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

Theorem pwtrrVD 34044
Description: Virtual deduction proof of pwtr 4690; see pwtrVD 34043 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 4534 . . 3  |-  ( Tr  A  <->  A. z A. y
( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
2 idn1 33764 . . . . . . . 8  |-  (. Tr  ~P A  ->.  Tr  ~P A ).
3 idn2 33812 . . . . . . . . 9  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  ( z  e.  y  /\  y  e.  A ) ).
4 simpr 459 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  A )  ->  y  e.  A )
53, 4e2 33830 . . . . . . . 8  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  e.  A ).
6 pwtrrVD.1 . . . . . . . . 9  |-  A  e. 
_V
76pwid 4013 . . . . . . . 8  |-  A  e. 
~P A
8 trel 4539 . . . . . . . . 9  |-  ( Tr 
~P A  ->  (
( y  e.  A  /\  A  e.  ~P A )  ->  y  e.  ~P A ) )
98expd 434 . . . . . . . 8  |-  ( Tr 
~P A  ->  (
y  e.  A  -> 
( A  e.  ~P A  ->  y  e.  ~P A ) ) )
102, 5, 7, 9e120 33862 . . . . . . 7  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  e.  ~P A ).
11 elpwi 4008 . . . . . . 7  |-  ( y  e.  ~P A  -> 
y  C_  A )
1210, 11e2 33830 . . . . . 6  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  y  C_  A ).
13 simpl 455 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  y )
143, 13e2 33830 . . . . . 6  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  z  e.  y ).
15 ssel 3483 . . . . . 6  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
1612, 14, 15e22 33870 . . . . 5  |-  (. Tr  ~P A ,. ( z  e.  y  /\  y  e.  A )  ->.  z  e.  A ).
1716in2 33804 . . . 4  |-  (. Tr  ~P A  ->.  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ).
1817gen12 33817 . . 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 33890 . 2  |-  (. Tr  ~P A  ->.  Tr  A ).
2120in1 33761 1  |-  ( Tr 
~P A  ->  Tr  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    e. wcel 1823   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   Tr wtr 4532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-in 3468  df-ss 3475  df-pw 4001  df-uni 4236  df-tr 4533  df-vd1 33760  df-vd2 33768
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator