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.

Step	Hyp	Ref	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 )
5	3, 4	e2 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 )
7	5, 6	e2 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 )
9	3, 8	e2 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 ) )
11	7, 9, 10	e22 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 ) )
13	2, 11, 12	e12 31291	. . . . . 6 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z C_ A ).
14		vex 2973	. . . . . . 7 |- z e. _V
15	14	elpw 3863	. . . . . 6 |- ( z e. ~P A <-> z C_ A )
16	13, 15	e2bir 31189	. . . . 5 |- (. Tr A ,. ( z e. y /\ y e. ~P A ) ->. z e. ~P A ).
17	16	in2 31161	. . . 4 |- (. Tr A ->. ( ( z e. y /\ y e. ~P A ) -> z e. ~P A ) ).
18	17	gen12 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 ) )
20	1, 18, 19	e01 31247	. 2 |- (. Tr A ->. Tr ~P A ).
21	20	in1 31117	1 |- ( Tr A -> Tr ~P A )

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)