Theorem trel 2048

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

Assertion

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

Proof of Theorem trel

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . . 6 |- (x = B -> (x e. C <-> B e. C))
2		eleq1 1149	. . . . . . 7 |- (x = B -> (x e. A <-> B e. A))
3	2	imbi2d 464	. . . . . 6 |- (x = B -> ((C e. A -> x e. A) <-> (C e. A -> B e. A)))
4	1, 3	imbi12d 474	. . . . 5 |- (x = B -> ((x e. C -> (C e. A -> x e. A)) <-> (B e. C -> (C e. A -> B e. A))))
5	4	imbi2d 464	. . . 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 1150	. . . . . . . . 9 |- (y = C -> (x e. y <-> x e. C))
7		eleq1 1149	. . . . . . . . . 10 |- (y = C -> (y e. A <-> C e. A))
8	7	imbi1d 465	. . . . . . . . 9 |- (y = C -> ((y e. A -> x e. A) <-> (C e. A -> x e. A)))
9	6, 8	imbi12d 474	. . . . . . . 8 |- (y = C -> ((x e. y -> (y e. A -> x e. A)) <-> (x e. C -> (C e. A -> x e. A))))
10	9	imbi2d 464	. . . . . . 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 2043	. . . . . . . . . 10 |- (Tr A <-> A.xA.y((x e. y /\ y e. A) -> x e. A))
12	11	biimp 133	. . . . . . . . 9 |- (Tr A -> A.xA.y((x e. y /\ y e. A) -> x e. A))
13	12	19.21bbi 743	. . . . . . . 8 |- (Tr A -> ((x e. y /\ y e. A) -> x e. A))
14	13	exp3a 292	. . . . . . 7 |- (Tr A -> (x e. y -> (y e. A -> x e. A)))
15	10, 14	vtoclg 1383	. . . . . 6 |- (C e. A -> (Tr A -> (x e. C -> (C e. A -> x e. A))))
16	15	com4l 39	. . . . 5 |- (Tr A -> (x e. C -> (C e. A -> (C e. A -> x e. A))))
17		pm2.43 57	. . . . 5 |- ((C e. A -> (C e. A -> x e. A)) -> (C e. A -> x e. A))
18	16, 17	syl6 23	. . . 4 |- (Tr A -> (x e. C -> (C e. A -> x e. A)))
19	5, 18	vtoclg 1383	. . 3 |- (B e. C -> (Tr A -> (B e. C -> (C e. A -> B e. A))))
20	19	pm2.43b 61	. 2 |- (Tr A -> (B e. C -> (C e. A -> B e. A)))
21	20	imp3a 279	1 |- (Tr A -> ((B e. C /\ C e. A) -> B e. A))

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672   e. wel 803   = wceq 1091   e. wcel 1092  Tr wtr 2041

This theorem is referenced by:  trel3 2049  ordn2lp 2219  ordelord 2221  tz7.7 2224  ordtr1 2256  trsuc 2308  ordom 2382  elnn 2383  zfregs 3491

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920  df-tr 2042

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