Theorem sspwtrALT 32575

Description: Virtual deduction proof of sspwtr 32574. This proof is the same as the proof of sspwtr 32574 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 4535	. . 3 |- ( Tr A <-> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
2		simpr 461	. . . . . 6 |- ( ( z e. y /\ y e. A ) -> y e. A )
3		ssel 3491	. . . . . 6 |- ( A C_ ~P A -> ( y e. A -> y e. ~P A ) )
4		elpwi 4012	. . . . . 6 |- ( y e. ~P A -> y C_ A )
5	2, 3, 4	syl56 34	. . . . 5 |- ( A C_ ~P A -> ( ( z e. y /\ y e. A ) -> y C_ A ) )
6		idd 24	. . . . . 6 |- ( A C_ ~P A -> ( ( z e. y /\ y e. A ) -> ( z e. y /\ y e. A ) ) )
7		simpl 457	. . . . . 6 |- ( ( z e. y /\ y e. A ) -> z e. y )
8	6, 7	syl6 33	. . . . 5 |- ( A C_ ~P A -> ( ( z e. y /\ y e. A ) -> z e. y ) )
9		ssel 3491	. . . . 5 |- ( y C_ A -> ( z e. y -> z e. A ) )
10	5, 8, 9	syl6c 64	. . . 4 |- ( A C_ ~P A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
11	10	alrimivv 1691	. . 3 |- ( A C_ ~P A -> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
12		bi2 198	. . 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 63	. 2 |- ( A C_ ~P A -> Tr A )
14	13	idiALT 32172	1 |- ( A C_ ~P A -> Tr A )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1372   e. wcel 1762   C_ wss 3469   ~Pcpw 4003   Tr wtr 4533

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 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-in 3476  df-ss 3483  df-pw 4005  df-uni 4239  df-tr 4534

This theorem is referenced by: (None)