Theorem pwtrVD 37260

Description: Virtual deduction proof of pwtr 4667; see pwtrrVD 37261 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 4513	. . 3 |- ( Tr ~P A <-> A. z A. y ( ( z e. y /\ y e. ~P A ) -> z e. ~P A ) )
2		idn1 36987	. . . . . . 7 |- (. Tr A ->. Tr A ).
3		idn2 37035	. . . . . . . . . 10 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. ( z e. y /\ y e. ~P A ) ).
4		simpr 467	. . . . . . . . . 10 |- ( ( z e. y /\ y e. ~P A ) -> y e. ~P A )
5	3, 4	e2 37053	. . . . . . . . 9 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. y e. ~P A ).
6		elpwi 3972	. . . . . . . . 9 |- ( y e. ~P A -> y C_ A )
7	5, 6	e2 37053	. . . . . . . 8 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. y C_ A ).
8		simpl 463	. . . . . . . . 9 |- ( ( z e. y /\ y e. ~P A ) -> z e. y )
9	3, 8	e2 37053	. . . . . . . 8 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. y ).
10		ssel 3438	. . . . . . . 8 |- ( y C_ A -> ( z e. y -> z e. A ) )
11	7, 9, 10	e22 37093	. . . . . . 7 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. A ).
12		trss 4520	. . . . . . 7 |- ( Tr A -> ( z e. A -> z C_ A ) )
13	2, 11, 12	e12 37151	. . . . . 6 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z C_ A ).
14		vex 3060	. . . . . . 7 |- z e. _V
15	14	elpw 3969	. . . . . 6 |- ( z e. ~P A <-> z C_ A )
16	13, 15	e2bir 37055	. . . . 5 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. ~P A ).
17	16	in2 37027	. . . 4 |- (. Tr A ->. ( ( z e. y /\ y e. ~P A ) -> z e. ~P A ) ).
18	17	gen12 37040	. . 3 |- (. Tr A ->. A. z A. y ( ( z e. y /\ y e. ~P A ) -> z e. ~P A ) ).
19		biimpr 203	. . 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 37113	. 2 |- (. Tr A ->. Tr ~P A ).
21	20	in1 36984	1 |- ( Tr A -> Tr ~P A )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1453   e. wcel 1898   C_ wss 3416   ~Pcpw 3963   Tr wtr 4511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-v 3059  df-in 3423  df-ss 3430  df-pw 3965  df-uni 4213  df-tr 4512  df-vd1 36983  df-vd2 36991

This theorem is referenced by: (None)