Theorem sspwtrALT 16644

Description: Virtual deduction proof of sspwtr 16643. This proof is the same as the proof of sspwtr 16643 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A class which is a subclass of its power class is transitive.

Assertion

Ref	Expression
sspwtrALT	|- (A C_ ~PA -> Tr A)

Proof of Theorem sspwtrALT

Step	Hyp	Ref	Expression
1		dftr2 3413	. . 3 |- (Tr A <-> A.zA.y((z e. y /\ y e. A) -> z e. A))
2		id 73	. . . . . . . 8 |- (A C_ ~PA -> A C_ ~PA)
3		idd 75	. . . . . . . . 9 |- (A C_ ~PA -> ((z e. y /\ y e. A) -> (z e. y /\ y e. A)))
4		simpr 350	. . . . . . . . 9 |- ((z e. y /\ y e. A) -> y e. A)
5	3, 4	syl6 25	. . . . . . . 8 |- (A C_ ~PA -> ((z e. y /\ y e. A) -> y e. A))
6		ssel 2615	. . . . . . . 8 |- (A C_ ~PA -> (y e. A -> y e. ~PA))
7	2, 5, 6	sylsyld 32	. . . . . . 7 |- (A C_ ~PA -> ((z e. y /\ y e. A) -> y e. ~PA))
8		elpwi 3039	. . . . . . 7 |- (y e. ~PA -> y C_ A)
9	7, 8	syl6 25	. . . . . 6 |- (A C_ ~PA -> ((z e. y /\ y e. A) -> y C_ A))
10		simpl 346	. . . . . . 7 |- ((z e. y /\ y e. A) -> z e. y)
11	3, 10	syl6 25	. . . . . 6 |- (A C_ ~PA -> ((z e. y /\ y e. A) -> z e. y))
12		ssel 2615	. . . . . 6 |- (y C_ A -> (z e. y -> z e. A))
13	9, 11, 12	ee22 1272	. . . . 5 |- (A C_ ~PA -> ((z e. y /\ y e. A) -> z e. A))
14	13	iin2 16507	. . . 4 |- (A C_ ~PA -> ((z e. y /\ y e. A) -> z e. A))
15	14	19.21aivv 1665	. . 3 |- (A C_ ~PA -> A.zA.y((z e. y /\ y e. A) -> z e. A))
16		bi2 166	. . 3 |- ((Tr A <-> A.zA.y((z e. y /\ y e. A) -> z e. A)) -> (A.zA.y((z e. y /\ y e. A) -> z e. A) -> Tr A))
17	1, 15, 16	ee01 16582	. 2 |- (A C_ ~PA -> Tr A)
18	17	iin1 16482	1 |- (A C_ ~PA -> Tr A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300   C_ wss 2593  ~Pcpw 3032  Tr wtr 3411

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-ss 2605  df-pw 3035  df-uni 3178  df-tr 3412

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