Theorem trsspwALT2 32990

Description: Virtual deduction proof of trsspwALT 32989. This proof is the same as the proof of trsspwALT 32989 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 3498	. . 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 4554	. . . . . . 7 |- ( Tr A -> ( x e. A -> x C_ A ) )
5	2, 3, 4	sylsyld 56	. . . . . 6 |- ( Tr A -> ( x e. A -> x C_ A ) )
6		vex 3121	. . . . . . 7 |- x e. _V
7	6	elpw 4021	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
8	5, 7	syl6ibr 227	. . . . 5 |- ( Tr A -> ( x e. A -> x e. ~P A ) )
9	8	idiALT 32590	. . . 4 |- ( Tr A -> ( x e. A -> x e. ~P A ) )
10	9	alrimiv 1695	. . 3 |- ( Tr A -> A. x ( x e. A -> x e. ~P A ) )
11		bi2 198	. . 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 63	. 2 |- ( Tr A -> A C_ ~P A )
13	12	idiALT 32590	1 |- ( Tr A -> A C_ ~P A )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1377   e. wcel 1767   C_ wss 3481   ~Pcpw 4015   Tr wtr 4545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-v 3120  df-in 3488  df-ss 3495  df-pw 4017  df-uni 4251  df-tr 4546

This theorem is referenced by: (None)