Theorem trel 4199

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 4194	. 2 |- ( Tr A <-> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
2		eleq12 2296	. . . . . 6 |- ( ( y = B /\ x = C ) -> ( y e. x <-> B e. C ) )
3		eleq1 2294	. . . . . . 7 |- ( x = C -> ( x e. A <-> C e. A ) )
4	3	adantl 277	. . . . . 6 |- ( ( y = B /\ x = C ) -> ( x e. A <-> C e. A ) )
5	2, 4	anbi12d 473	. . . . 5 |- ( ( y = B /\ x = C ) -> ( ( y e. x /\ x e. A ) <-> ( B e. C /\ C e. A ) ) )
6		eleq1 2294	. . . . . 6 |- ( y = B -> ( y e. A <-> B e. A ) )
7	6	adantr 276	. . . . 5 |- ( ( y = B /\ x = C ) -> ( y e. A <-> B e. A ) )
8	5, 7	imbi12d 234	. . . 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 2898	. . 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 121	1 |- ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1396   = wceq 1398   e. wcel 2202   Tr wtr 4192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-uni 3899  df-tr 4193

This theorem is referenced by:  trel3  4200  ordtr1  4491  suctr  4524  trsuc  4525  ordn2lp  4649