Theorem sspwtrALT 39049

Description: Virtual deduction proof of sspwtr 39048. This proof is the same as the proof of sspwtr 39048 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 4754	. . 3 |- ( Tr A <-> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
2		simpr 477	. . . . . 6 |- ( ( z e. y /\ y e. A ) -> y e. A )
3		ssel 3597	. . . . . 6 |- ( A C_ ~P A -> ( y e. A -> y e. ~P A ) )
4		elpwi 4168	. . . . . 6 |- ( y e. ~P A -> y C_ A )
5	2, 3, 4	syl56 36	. . . . 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 473	. . . . . 6 |- ( ( z e. y /\ y e. A ) -> z e. y )
8	6, 7	syl6 35	. . . . 5 |- ( A C_ ~P A -> ( ( z e. y /\ y e. A ) -> z e. y ) )
9		ssel 3597	. . . . 5 |- ( y C_ A -> ( z e. y -> z e. A ) )
10	5, 8, 9	syl6c 70	. . . 4 |- ( A C_ ~P A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
11	10	alrimivv 1856	. . 3 |- ( A C_ ~P A -> A. z A. y ( ( z e. y /\ y e. A ) -> z e. A ) )
12		biimpr 210	. . 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 68	. 2 |- ( A C_ ~P A -> Tr A )
14	13	idiALT 38683	1 |- ( A C_ ~P A -> Tr A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-tr 4753

This theorem is referenced by: (None)