Theorem sspwtrALT 37210

Description: Virtual deduction proof of sspwtr 37209. This proof is the same as the proof of sspwtr 37209 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A class which is a subclass of its power class is transitive. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sspwtrALT	|- ( A C_ ~P A -> Tr A )

Proof of Theorem sspwtrALT

Dummy variables z y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr2 4499	. . 3 |- ( Tr A <-> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
2		simpr 463	. . . . . 6 |- ( ( z e. y /\ y e. A ) -> y e. A )
3		ssel 3426	. . . . . 6 |- ( A C_ ~P A -> ( y e. A -> y e. ~P A ) )
4		elpwi 3960	. . . . . 6 |- ( y e. ~P A -> y C_ A )
5	2, 3, 4	syl56 35	. . . . 5 |- ( A C_ ~P A -> ( ( z e. y /\ y e. A ) -> y C_ A ) )
6		idd 25	. . . . . 6 |- ( A C_ ~P A -> ( ( z e. y /\ y e. A ) -> ( z e. y /\ y e. A ) ) )
7		simpl 459	. . . . . 6 |- ( ( z e. y /\ y e. A ) -> z e. y )
8	6, 7	syl6 34	. . . . 5 |- ( A C_ ~P A -> ( ( z e. y /\ y e. A ) -> z e. y ) )
9		ssel 3426	. . . . 5 |- ( y C_ A -> ( z e. y -> z e. A ) )
10	5, 8, 9	syl6c 66	. . . 4 |- ( A C_ ~P A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
11	10	alrimivv 1774	. . 3 |- ( A C_ ~P A -> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
12		biimpr 202	. . 3 |- ( ( Tr A <-> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) ) -> ( A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) -> Tr A ) )
13	1, 11, 12	mpsyl 65	. 2 |- ( A C_ ~P A -> Tr A )
14	13	idiALT 36832	1 |- ( A C_ ~P A -> Tr A )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1442   e. wcel 1887   C_ wss 3404   ~Pcpw 3951   Tr wtr 4497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-in 3411  df-ss 3418  df-pw 3953  df-uni 4199  df-tr 4498

This theorem is referenced by: (None)