Theorem trel 2687

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 1534	. . . . . 6 |- (x = B -> (x e. C <-> B e. C))
2		eleq1 1534	. . . . . . 7 |- (x = B -> (x e. A <-> B e. A))
3	2	imbi2d 612	. . . . . 6 |- (x = B -> ((C e. A -> x e. A) <-> (C e. A -> B e. A)))
4	1, 3	imbi12d 626	. . . . 5 |- (x = B -> ((x e. C -> (C e. A -> x e. A)) <-> (B e. C -> (C e. A -> B e. A))))
5	4	imbi2d 612	. . . 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 1535	. . . . . . . . 9 |- (y = C -> (x e. y <-> x e. C))
7		eleq1 1534	. . . . . . . . . 10 |- (y = C -> (y e. A <-> C e. A))
8	7	imbi1d 613	. . . . . . . . 9 |- (y = C -> ((y e. A -> x e. A) <-> (C e. A -> x e. A)))
9	6, 8	imbi12d 626	. . . . . . . 8 |- (y = C -> ((x e. y -> (y e. A -> x e. A)) <-> (x e. C -> (C e. A -> x e. A))))
10	9	imbi2d 612	. . . . . . 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 2682	. . . . . . . . . 10 |- (Tr A <-> A.xA.y((x e. y /\ y e. A) -> x e. A))
12	11	biimp 151	. . . . . . . . 9 |- (Tr A -> A.xA.y((x e. y /\ y e. A) -> x e. A))
13	12	19.21bbi 1061	. . . . . . . 8 |- (Tr A -> ((x e. y /\ y e. A) -> x e. A))
14	13	exp3a 375	. . . . . . 7 |- (Tr A -> (x e. y -> (y e. A -> x e. A)))
15	10, 14	vtoclg 1847	. . . . . 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 63	. . . . 5 |- ((C e. A -> (C e. A -> x e. A)) -> (C e. A -> x e. A))
18	16, 17	syl6 22	. . . 4 |- (Tr A -> (x e. C -> (C e. A -> x e. A)))
19	5, 18	vtoclg 1847	. . 3 |- (B e. C -> (Tr A -> (B e. C -> (C e. A -> B e. A))))
20	19	pm2.43b 67	. 2 |- (Tr A -> (B e. C -> (C e. A -> B e. A)))
21	20	imp3a 361	1 |- (Tr A -> ((B e. C /\ C e. A) -> B e. A))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  Tr wtr 2680

This theorem is referenced by:  trel3 2688  ordn2lp 2968  ordelord 2970  tz7.7 2973  ordtr1 3001  trsuc 3055  ordom 3141  elnn 3142  zfregs 4647

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-uni 2504  df-tr 2681

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