Theorem sspwtrALT2 16645

Description: Short predicate calculus proof of the right-to-left implication of dftr4 3416. A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT 16644, which is the virtual deduction proof sspwtr 16643 without virtual deductions.

Assertion

Ref	Expression
sspwtrALT2	|- (A C_ ~PA -> Tr A)

Proof of Theorem sspwtrALT2

Step	Hyp	Ref	Expression
1		ssel 2615	. . . . . 6 |- (A C_ ~PA -> (y e. A -> y e. ~PA))
2	1	adantld 426	. . . . 5 |- (A C_ ~PA -> ((z e. y /\ y e. A) -> y e. ~PA))
3		elpwi 3039	. . . . 5 |- (y e. ~PA -> y C_ A)
4	2, 3	syl6 25	. . . 4 |- (A C_ ~PA -> ((z e. y /\ y e. A) -> y C_ A))
5		simpl 346	. . . . 5 |- ((z e. y /\ y e. A) -> z e. y)
6	5	a1i 8	. . . 4 |- (A C_ ~PA -> ((z e. y /\ y e. A) -> z e. y))
7		ssel 2615	. . . 4 |- (y C_ A -> (z e. y -> z e. A))
8	4, 6, 7	ee22 1272	. . 3 |- (A C_ ~PA -> ((z e. y /\ y e. A) -> z e. A))
9	8	19.21aivv 1665	. 2 |- (A C_ ~PA -> A.zA.y((z e. y /\ y e. A) -> z e. A))
10		dftr2 3413	. 2 |- (Tr A <-> A.zA.y((z e. y /\ y e. A) -> z e. A))
11	9, 10	sylibr 217	1 |- (A C_ ~PA -> Tr A)

Syntax hints:   -> wi 3   /\ 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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