Theorem trel 4087

Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof 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

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr2 4082	. 2 |- ( Tr A <-> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
2		eleq12 2231	. . . . . 6 |- ( ( y = B /\ x = C ) -> ( y e. x <-> B e. C ) )
3		eleq1 2229	. . . . . . 7 |- ( x = C -> ( x e. A <-> C e. A ) )
4	3	adantl 275	. . . . . 6 |- ( ( y = B /\ x = C ) -> ( x e. A <-> C e. A ) )
5	2, 4	anbi12d 465	. . . . 5 |- ( ( y = B /\ x = C ) -> ( ( y e. x /\ x e. A ) <-> ( B e. C /\ C e. A ) ) )
6		eleq1 2229	. . . . . 6 |- ( y = B -> ( y e. A <-> B e. A ) )
7	6	adantr 274	. . . . 5 |- ( ( y = B /\ x = C ) -> ( y e. A <-> B e. A ) )
8	5, 7	imbi12d 233	. . . 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	spc2gv 2817	. . 3 |- ( ( B e. C /\ C e. A ) -> ( A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) -> ( ( B e. C /\ C e. A ) -> B e. A ) ) )
10	9	pm2.43b 52	. 2 |- ( A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) -> ( ( B e. C /\ C e. A ) -> B e. A ) )
11	1, 10	sylbi 120	1 |- ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   e. wcel 2136   Tr wtr 4080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081

This theorem is referenced by:  trel3  4088  ordtr1  4366  suctr  4399  trsuc  4400  ordn2lp  4522