Theorem sspwtrALT 31858

Description: Virtual deduction proof of sspwtr 31857. This proof is the same as the proof of sspwtr 31857 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 4487	. . 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 3450	. . . . . 6 |- ( A C_ ~P A -> ( y e. A -> y e. ~P A ) )
4		elpwi 3969	. . . . . 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 3450	. . . . 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 1687	. . 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 31455	1 |- ( A C_ ~P A -> Tr A )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1368   e. wcel 1758   C_ wss 3428   ~Pcpw 3960   Tr wtr 4485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3072  df-in 3435  df-ss 3442  df-pw 3962  df-uni 4192  df-tr 4486

This theorem is referenced by: (None)