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Theorem pwtrVD 31394
Description: Virtual deduction proof of pwtr 4542; see pwtrrVD 31395 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 4384 . . 3  |-  ( Tr 
~P A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  ~P A )  -> 
z  e.  ~P A
) )
2 idn1 31120 . . . . . . 7  |-  (. Tr  A 
->.  Tr  A ).
3 idn2 31169 . . . . . . . . . 10  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  ( z  e.  y  /\  y  e.  ~P A ) ).
4 simpr 458 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  ~P A
)  ->  y  e.  ~P A )
53, 4e2 31187 . . . . . . . . 9  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  y  e.  ~P A ).
6 elpwi 3866 . . . . . . . . 9  |-  ( y  e.  ~P A  -> 
y  C_  A )
75, 6e2 31187 . . . . . . . 8  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  y  C_  A ).
8 simpl 454 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  ~P A
)  ->  z  e.  y )
93, 8e2 31187 . . . . . . . 8  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  y ).
10 ssel 3347 . . . . . . . 8  |-  ( y 
C_  A  ->  (
z  e.  y  -> 
z  e.  A ) )
117, 9, 10e22 31227 . . . . . . 7  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  A ).
12 trss 4391 . . . . . . 7  |-  ( Tr  A  ->  ( z  e.  A  ->  z  C_  A ) )
132, 11, 12e12 31291 . . . . . 6  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  C_  A ).
14 vex 2973 . . . . . . 7  |-  z  e. 
_V
1514elpw 3863 . . . . . 6  |-  ( z  e.  ~P A  <->  z  C_  A )
1613, 15e2bir 31189 . . . . 5  |-  (. Tr  A ,. ( z  e.  y  /\  y  e. 
~P A )  ->.  z  e.  ~P A ).
1716in2 31161 . . . 4  |-  (. Tr  A 
->.  ( ( z  e.  y  /\  y  e. 
~P A )  -> 
z  e.  ~P A
) ).
1817gen12 31174 . . 3  |-  (. Tr  A 
->.  A. z A. y
( ( z  e.  y  /\  y  e. 
~P A )  -> 
z  e.  ~P A
) ).
19 bi2 198 . . 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 31247 . 2  |-  (. Tr  A 
->.  Tr  ~P A ).
2120in1 31117 1  |-  ( Tr  A  ->  Tr  ~P A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1362    e. wcel 1761    C_ wss 3325   ~Pcpw 3857   Tr wtr 4382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-v 2972  df-in 3332  df-ss 3339  df-pw 3859  df-uni 4089  df-tr 4383  df-vd1 31116  df-vd2 31125
This theorem is referenced by: (None)
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