Theorem pwtrVD 32704

Description: Virtual deduction proof of pwtr 4700; see pwtrrVD 32705 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.

Step	Hyp	Ref	Expression
1		dftr2 4542	. . 3 |- ( Tr ~P A <-> A. z A. y ( ( z e. y /\ y e. ~P A ) -> z e. ~P A ) )
2		idn1 32431	. . . . . . 7 |- (. Tr A ->. Tr A ).
3		idn2 32479	. . . . . . . . . 10 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. ( z e. y /\ y e. ~P A ) ).
4		simpr 461	. . . . . . . . . 10 |- ( ( z e. y /\ y e. ~P A ) -> y e. ~P A )
5	3, 4	e2 32497	. . . . . . . . 9 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. y e. ~P A ).
6		elpwi 4019	. . . . . . . . 9 |- ( y e. ~P A -> y C_ A )
7	5, 6	e2 32497	. . . . . . . 8 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. y C_ A ).
8		simpl 457	. . . . . . . . 9 |- ( ( z e. y /\ y e. ~P A ) -> z e. y )
9	3, 8	e2 32497	. . . . . . . 8 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. y ).
10		ssel 3498	. . . . . . . 8 |- ( y C_ A -> ( z e. y -> z e. A ) )
11	7, 9, 10	e22 32537	. . . . . . 7 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. A ).
12		trss 4549	. . . . . . 7 |- ( Tr A -> ( z e. A -> z C_ A ) )
13	2, 11, 12	e12 32601	. . . . . 6 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z C_ A ).
14		vex 3116	. . . . . . 7 |- z e. _V
15	14	elpw 4016	. . . . . 6 |- ( z e. ~P A <-> z C_ A )
16	13, 15	e2bir 32499	. . . . 5 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. ~P A ).
17	16	in2 32471	. . . 4 |- (. Tr A ->. ( ( z e. y /\ y e. ~P A ) -> z e. ~P A ) ).
18	17	gen12 32484	. . 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 ) )
20	1, 18, 19	e01 32557	. 2 |- (. Tr A ->. Tr ~P A ).
21	20	in1 32428	1 |- ( Tr A -> Tr ~P A )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1377   e. wcel 1767   C_ wss 3476   ~Pcpw 4010   Tr wtr 4540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-in 3483  df-ss 3490  df-pw 4012  df-uni 4246  df-tr 4541  df-vd1 32427  df-vd2 32435

This theorem is referenced by: (None)