Theorem sspwtrALT 37071

Description: Virtual deduction proof of sspwtr 37070. This proof is the same as the proof of sspwtr 37070 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 4517	. . 3 |- ( Tr A <-> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
2		simpr 462	. . . . . 6 |- ( ( z e. y /\ y e. A ) -> y e. A )
3		ssel 3458	. . . . . 6 |- ( A C_ ~P A -> ( y e. A -> y e. ~P A ) )
4		elpwi 3988	. . . . . 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 458	. . . . . 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 3458	. . . . 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 1764	. . 3 |- ( A C_ ~P A -> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
12		biimpr 201	. . 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 36690	1 |- ( A C_ ~P A -> Tr A )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   A.wal 1435   e. wcel 1868   C_ wss 3436   ~Pcpw 3979   Tr wtr 4515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-in 3443  df-ss 3450  df-pw 3981  df-uni 4217  df-tr 4516

This theorem is referenced by: (None)