Theorem pwtrVD 39059

Description: Virtual deduction proof of pwtr 4921; see pwtrrVD 39060 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 4754	. . 3 |- ( Tr ~P A <-> A. z A. y ( ( z e. y /\ y e. ~P A ) -> z e. ~P A ) )
2		idn1 38790	. . . . . . 7 |- (. Tr A ->. Tr A ).
3		idn2 38838	. . . . . . . . . 10 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. ( z e. y /\ y e. ~P A ) ).
4		simpr 477	. . . . . . . . . 10 |- ( ( z e. y /\ y e. ~P A ) -> y e. ~P A )
5	3, 4	e2 38856	. . . . . . . . 9 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. y e. ~P A ).
6		elpwi 4168	. . . . . . . . 9 |- ( y e. ~P A -> y C_ A )
7	5, 6	e2 38856	. . . . . . . 8 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. y C_ A ).
8		simpl 473	. . . . . . . . 9 |- ( ( z e. y /\ y e. ~P A ) -> z e. y )
9	3, 8	e2 38856	. . . . . . . 8 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. y ).
10		ssel 3597	. . . . . . . 8 |- ( y C_ A -> ( z e. y -> z e. A ) )
11	7, 9, 10	e22 38896	. . . . . . 7 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. A ).
12		trss 4761	. . . . . . 7 |- ( Tr A -> ( z e. A -> z C_ A ) )
13	2, 11, 12	e12 38951	. . . . . 6 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z C_ A ).
14		vex 3203	. . . . . . 7 |- z e. _V
15	14	elpw 4164	. . . . . 6 |- ( z e. ~P A <-> z C_ A )
16	13, 15	e2bir 38858	. . . . 5 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. ~P A ).
17	16	in2 38830	. . . 4 |- (. Tr A ->. ( ( z e. y /\ y e. ~P A ) -> z e. ~P A ) ).
18	17	gen12 38843	. . 3 |- (. Tr A ->. A. z A. y ( ( z e. y /\ y e. ~P A ) -> z e. ~P A ) ).
19		biimpr 210	. . 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 38916	. 2 |- (. Tr A ->. Tr ~P A ).
21	20	in1 38787	1 |- ( Tr A -> Tr ~P A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-tr 4753  df-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)