Theorem trsspwALT2 39046

Description: Virtual deduction proof of trsspwALT 39045. This proof is the same as the proof of trsspwALT 39045 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 3591	. . 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 4761	. . . . . . 7 |- ( Tr A -> ( x e. A -> x C_ A ) )
5	2, 3, 4	sylsyld 61	. . . . . 6 |- ( Tr A -> ( x e. A -> x C_ A ) )
6		vex 3203	. . . . . . 7 |- x e. _V
7	6	elpw 4164	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
8	5, 7	syl6ibr 242	. . . . 5 |- ( Tr A -> ( x e. A -> x e. ~P A ) )
9	8	idiALT 38683	. . . 4 |- ( Tr A -> ( x e. A -> x e. ~P A ) )
10	9	alrimiv 1855	. . 3 |- ( Tr A -> A. x ( x e. A -> x e. ~P A ) )
11		biimpr 210	. . 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 68	. 2 |- ( Tr A -> A C_ ~P A )
13	12	idiALT 38683	1 |- ( Tr A -> A C_ ~P A )

Syntax hints:   -> wi 4   <-> wb 196   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-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-tr 4753

This theorem is referenced by: (None)