Theorem trsspwALT2 31558

Description: Virtual deduction proof of trsspwALT 31557. This proof is the same as the proof of trsspwALT 31557 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 3350	. . 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 4399	. . . . . . 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 2980	. . . . . . 7 |- x e. _V
7	6	elpw 3871	. . . . . 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 31158	. . . 4 |- ( Tr A -> ( x e. A -> x e. ~P A ) )
10	9	alrimiv 1685	. . 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 31158	1 |- ( Tr A -> A C_ ~P A )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1367   e. wcel 1756   C_ wss 3333   ~Pcpw 3865   Tr wtr 4390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-v 2979  df-in 3340  df-ss 3347  df-pw 3867  df-uni 4097  df-tr 4391

This theorem is referenced by: (None)