Theorem trel 3418

Description: In a transitive class, the membership relation is transitive. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

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

Proof of Theorem trel

Step	Hyp	Ref	Expression
1		dftr2 3413	. 2 |- (Tr A <-> A.yA.x((y e. x /\ x e. A) -> y e. A))
2		eleq12 1959	. . . . . 6 |- ((y = B /\ x = C) -> (y e. x <-> B e. C))
3		eleq1 1957	. . . . . . 7 |- (x = C -> (x e. A <-> C e. A))
4	3	adantl 424	. . . . . 6 |- ((y = B /\ x = C) -> (x e. A <-> C e. A))
5	2, 4	anbi12d 690	. . . . 5 |- ((y = B /\ x = C) -> ((y e. x /\ x e. A) <-> (B e. C /\ C e. A)))
6		eleq1 1957	. . . . . 6 |- (y = B -> (y e. A <-> B e. A))
7	6	adantr 425	. . . . 5 |- ((y = B /\ x = C) -> (y e. A <-> B e. A))
8	5, 7	imbi12d 688	. . . 4 |- ((y = B /\ x = C) -> (((y e. x /\ x e. A) -> y e. A) <-> ((B e. C /\ C e. A) -> B e. A)))
9	8	cla42gv 2367	. . 3 |- ((B e. C /\ C e. A) -> (A.yA.x((y e. x /\ x e. A) -> y e. A) -> ((B e. C /\ C e. A) -> B e. A)))
10	9	pm2.43b 81	. 2 |- (A.yA.x((y e. x /\ x e. A) -> y e. A) -> ((B e. C /\ C e. A) -> B e. A))
11	1, 10	sylbi 216	1 |- (Tr A -> ((B e. C /\ C e. A) -> B e. A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  Tr wtr 3411

This theorem is referenced by:  trel3 3420  trintss 3427  ordn2lp 3678  ordelord 3680  tz7.7 3684  ordtr1 3707  suctr 3751  trsuc 3752  trsucOLD 3753  trsuc2 3754  ordom 3960  ordomOLD 3961  elnn 3962  zfregs 5754  tratrb 5831  truniALT 5845  trsuc2OLD 13793  trintssOLD 13795  dfon2lem6 13854  intrtael 15256  fnctartar 15284  pwtrrVD 16648  pwtrr 16649  suctrALT2VD 16660  suctrALT2 16661  tratrbVD 16685  truniALTVD 16702  trintALTVD 16704  trintALT 16705

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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