Theorem pwtr 16647

Description: The power class of a transitive class is transitive. The proof of this theorem was automatically generated from pwtrVD 16646 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it.

Assertion

Ref	Expression
pwtr	|- (Tr A -> Tr ~PA)

Proof of Theorem pwtr

Step	Hyp	Ref	Expression
1		simpr 350	. . . . . . . 8 |- ((z e. y /\ y e. ~PA) -> y e. ~PA)
2	1	a1i 8	. . . . . . 7 |- (Tr A -> ((z e. y /\ y e. ~PA) -> y e. ~PA))
3		elpwi 3039	. . . . . . 7 |- (y e. ~PA -> y C_ A)
4	2, 3	syl6 25	. . . . . 6 |- (Tr A -> ((z e. y /\ y e. ~PA) -> y C_ A))
5		simpl 346	. . . . . . 7 |- ((z e. y /\ y e. ~PA) -> z e. y)
6	5	a1i 8	. . . . . 6 |- (Tr A -> ((z e. y /\ y e. ~PA) -> z e. y))
7		ssel 2615	. . . . . 6 |- (y C_ A -> (z e. y -> z e. A))
8	4, 6, 7	ee22 1272	. . . . 5 |- (Tr A -> ((z e. y /\ y e. ~PA) -> z e. A))
9		trss 3421	. . . . 5 |- (Tr A -> (z e. A -> z C_ A))
10	8, 9	syld 30	. . . 4 |- (Tr A -> ((z e. y /\ y e. ~PA) -> z C_ A))
11		visset 2295	. . . . 5 |- z e. _V
12	11	elpw 3037	. . . 4 |- (z e. ~PA <-> z C_ A)
13	10, 12	syl6ibr 230	. . 3 |- (Tr A -> ((z e. y /\ y e. ~PA) -> z e. ~PA))
14	13	19.21aivv 1665	. 2 |- (Tr A -> A.zA.y((z e. y /\ y e. ~PA) -> z e. ~PA))
15		dftr2 3413	. 2 |- (Tr ~PA <-> A.zA.y((z e. y /\ y e. ~PA) -> z e. ~PA))
16	14, 15	sylibr 217	1 |- (Tr A -> Tr ~PA)

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-ral 2109  df-v 2294  df-in 2603  df-ss 2605  df-pw 3035  df-uni 3178  df-tr 3412

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