Theorem pwtrr 16649

Description: A set is transitive if its power set is. The proof of this theorem was automatically generated from pwtrrVD 16648 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it.

Hypothesis

Ref	Expression
pwtrr.1	|- A e. _V

Assertion

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

Proof of Theorem pwtrr

Step	Hyp	Ref	Expression
1		id 73	. . . . . 6 |- (Tr ~PA -> Tr ~PA)
2		simpr 350	. . . . . . 7 |- ((z e. y /\ y e. A) -> y e. A)
3	2	a1i 8	. . . . . 6 |- (Tr ~PA -> ((z e. y /\ y e. A) -> y e. A))
4		pwtrr.1	. . . . . . 7 |- A e. _V
5	4	pwid 3042	. . . . . 6 |- A e. ~PA
6		trel 3418	. . . . . . 7 |- (Tr ~PA -> ((y e. A /\ A e. ~PA) -> y e. ~PA))
7	6	exp3a 405	. . . . . 6 |- (Tr ~PA -> (y e. A -> (A e. ~PA -> y e. ~PA)))
8	1, 3, 5, 7	ee120 16554	. . . . 5 |- (Tr ~PA -> ((z e. y /\ y e. A) -> y e. ~PA))
9		elpwi 3039	. . . . 5 |- (y e. ~PA -> y C_ A)
10	8, 9	syl6 25	. . . 4 |- (Tr ~PA -> ((z e. y /\ y e. A) -> y C_ A))
11		simpl 346	. . . . 5 |- ((z e. y /\ y e. A) -> z e. y)
12	11	a1i 8	. . . 4 |- (Tr ~PA -> ((z e. y /\ y e. A) -> z e. y))
13		ssel 2615	. . . 4 |- (y C_ A -> (z e. y -> z e. A))
14	10, 12, 13	ee22 1272	. . 3 |- (Tr ~PA -> ((z e. y /\ y e. A) -> z e. A))
15	14	19.21aivv 1665	. 2 |- (Tr ~PA -> A.zA.y((z e. y /\ y e. A) -> z e. A))
16		dftr2 3413	. 2 |- (Tr A <-> A.zA.y((z e. y /\ y e. A) -> z e. A))
17	15, 16	sylibr 217	1 |- (Tr ~PA -> Tr A)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   e. wcel 1300  _Vcvv 2292   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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