Theorem trsspwALT2 33325

Description: Virtual deduction proof of trsspwALT 33324. This proof is the same as the proof of trsspwALT 33324 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 3475	. . 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 4535	. . . . . . 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 3096	. . . . . . 7 |- x e. _V
7	6	elpw 3999	. . . . . 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 32926	. . . 4 |- ( Tr A -> ( x e. A -> x e. ~P A ) )
10	9	alrimiv 1704	. . 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 32926	1 |- ( Tr A -> A C_ ~P A )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1379   e. wcel 1802   C_ wss 3458   ~Pcpw 3993   Tr wtr 4526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ral 2796  df-v 3095  df-in 3465  df-ss 3472  df-pw 3995  df-uni 4231  df-tr 4527

This theorem is referenced by: (None)