Theorem trelOLD 3419

Description: In a transitive class, the membership relation is transitive.

Assertion

Ref	Expression
trelOLD	|- (Tr A -> ((B e. C /\ C e. A) -> B e. A))

Proof of Theorem trelOLD

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . . 6 |- (x = B -> (x e. C <-> B e. C))
2		eleq1 1957	. . . . . . 7 |- (x = B -> (x e. A <-> B e. A))
3	2	imbi2d 674	. . . . . 6 |- (x = B -> ((C e. A -> x e. A) <-> (C e. A -> B e. A)))
4	1, 3	imbi12d 688	. . . . 5 |- (x = B -> ((x e. C -> (C e. A -> x e. A)) <-> (B e. C -> (C e. A -> B e. A))))
5	4	imbi2d 674	. . . 4 |- (x = B -> ((Tr A -> (x e. C -> (C e. A -> x e. A))) <-> (Tr A -> (B e. C -> (C e. A -> B e. A)))))
6		eleq2 1958	. . . . . . . . 9 |- (y = C -> (x e. y <-> x e. C))
7		eleq1 1957	. . . . . . . . . 10 |- (y = C -> (y e. A <-> C e. A))
8	7	imbi1d 675	. . . . . . . . 9 |- (y = C -> ((y e. A -> x e. A) <-> (C e. A -> x e. A)))
9	6, 8	imbi12d 688	. . . . . . . 8 |- (y = C -> ((x e. y -> (y e. A -> x e. A)) <-> (x e. C -> (C e. A -> x e. A))))
10	9	imbi2d 674	. . . . . . 7 |- (y = C -> ((Tr A -> (x e. y -> (y e. A -> x e. A))) <-> (Tr A -> (x e. C -> (C e. A -> x e. A)))))
11		dftr2 3413	. . . . . . . . . 10 |- (Tr A <-> A.xA.y((x e. y /\ y e. A) -> x e. A))
12	11	biimpi 168	. . . . . . . . 9 |- (Tr A -> A.xA.y((x e. y /\ y e. A) -> x e. A))
13	12	19.21bbi 1409	. . . . . . . 8 |- (Tr A -> ((x e. y /\ y e. A) -> x e. A))
14	13	exp3a 405	. . . . . . 7 |- (Tr A -> (x e. y -> (y e. A -> x e. A)))
15	10, 14	vtoclg 2346	. . . . . 6 |- (C e. A -> (Tr A -> (x e. C -> (C e. A -> x e. A))))
16	15	com4l 43	. . . . 5 |- (Tr A -> (x e. C -> (C e. A -> (C e. A -> x e. A))))
17		pm2.43 77	. . . . 5 |- ((C e. A -> (C e. A -> x e. A)) -> (C e. A -> x e. A))
18	16, 17	syl6 25	. . . 4 |- (Tr A -> (x e. C -> (C e. A -> x e. A)))
19	5, 18	vtoclg 2346	. . 3 |- (B e. C -> (Tr A -> (B e. C -> (C e. A -> B e. A))))
20	19	pm2.43b 81	. 2 |- (Tr A -> (B e. C -> (C e. A -> B e. A)))
21	20	imp3a 388	1 |- (Tr A -> ((B e. C /\ C e. A) -> B e. A))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  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-uni 3178  df-tr 3412

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