Theorem trsspwALT2 37270

Description: Virtual deduction proof of trsspwALT 37269. This proof is the same as the proof of trsspwALT 37269 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A transitive class is a subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
trsspwALT2	|- ( Tr A -> A C_ ~P A )

Proof of Theorem trsspwALT2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3407	. . 3 |- ( A C_ ~P A <-> A. x ( x e. A -> x e. ~P A ) )
2		id 22	. . . . . . 7 |- ( Tr A -> Tr A )
3		idd 24	. . . . . . 7 |- ( Tr A -> ( x e. A -> x e. A ) )
4		trss 4499	. . . . . . 7 |- ( Tr A -> ( x e. A -> x C_ A ) )
5	2, 3, 4	sylsyld 57	. . . . . 6 |- ( Tr A -> ( x e. A -> x C_ A ) )
6		vex 3034	. . . . . . 7 |- x e. _V
7	6	elpw 3948	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
8	5, 7	syl6ibr 235	. . . . 5 |- ( Tr A -> ( x e. A -> x e. ~P A ) )
9	8	idiALT 36902	. . . 4 |- ( Tr A -> ( x e. A -> x e. ~P A ) )
10	9	alrimiv 1781	. . 3 |- ( Tr A -> A. x ( x e. A -> x e. ~P A ) )
11		biimpr 203	. . 3 |- ( ( A C_ ~P A <-> A. x ( x e. A -> x e. ~P A ) ) -> ( A. x ( x e. A -> x e. ~P A ) -> A C_ ~P A ) )
12	1, 10, 11	mpsyl 64	. 2 |- ( Tr A -> A C_ ~P A )
13	12	idiALT 36902	1 |- ( Tr A -> A C_ ~P A )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1450   e. wcel 1904   C_ wss 3390   ~Pcpw 3942   Tr wtr 4490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-v 3033  df-in 3397  df-ss 3404  df-pw 3944  df-uni 4191  df-tr 4491

This theorem is referenced by: (None)