Theorem pwtrVD 31565

Description: Virtual deduction proof of pwtr 4550; see pwtrrVD 31566 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 4392	. . 3 |- ( Tr ~P A <-> A. z A. y ( ( z e. y /\ y e. ~P A ) -> z e. ~P A ) )
2		idn1 31292	. . . . . . 7 |- (. Tr A ->. Tr A ).
3		idn2 31340	. . . . . . . . . 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 31358	. . . . . . . . 9 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. y e. ~P A ).
6		elpwi 3874	. . . . . . . . 9 |- ( y e. ~P A -> y C_ A )
7	5, 6	e2 31358	. . . . . . . 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 31358	. . . . . . . 8 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. y ).
10		ssel 3355	. . . . . . . 8 |- ( y C_ A -> ( z e. y -> z e. A ) )
11	7, 9, 10	e22 31398	. . . . . . 7 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. A ).
12		trss 4399	. . . . . . 7 |- ( Tr A -> ( z e. A -> z C_ A ) )
13	2, 11, 12	e12 31462	. . . . . 6 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z C_ A ).
14		vex 2980	. . . . . . 7 |- z e. _V
15	14	elpw 3871	. . . . . 6 |- ( z e. ~P A <-> z C_ A )
16	13, 15	e2bir 31360	. . . . 5 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. ~P A ).
17	16	in2 31332	. . . 4 |- (. Tr A ->. ( ( z e. y /\ y e. ~P A ) -> z e. ~P A ) ).
18	17	gen12 31345	. . 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 31418	. 2 |- (. Tr A ->. Tr ~P A ).
21	20	in1 31289	1 |- ( Tr A -> Tr ~P A )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1367   e. wcel 1756   C_ wss 3333   ~Pcpw 3865   Tr wtr 4390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-v 2979  df-in 3340  df-ss 3347  df-pw 3867  df-uni 4097  df-tr 4391  df-vd1 31288  df-vd2 31296

This theorem is referenced by: (None)