Theorem trel 3861

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 3856	. 2 |- ( Tr A <-> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
2		eleq12 2102	. . . . . 6 |- ( ( y = B /\ x = C ) -> ( y e. x <-> B e. C ) )
3		eleq1 2100	. . . . . . 7 |- ( x = C -> ( x e. A <-> C e. A ) )
4	3	adantl 262	. . . . . 6 |- ( ( y = B /\ x = C ) -> ( x e. A <-> C e. A ) )
5	2, 4	anbi12d 442	. . . . 5 |- ( ( y = B /\ x = C ) -> ( ( y e. x /\ x e. A ) <-> ( B e. C /\ C e. A ) ) )
6		eleq1 2100	. . . . . 6 |- ( y = B -> ( y e. A <-> B e. A ) )
7	6	adantr 261	. . . . 5 |- ( ( y = B /\ x = C ) -> ( y e. A <-> B e. A ) )
8	5, 7	imbi12d 223	. . . 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 2643	. . 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 46	. 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 114	1 |- ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   Tr wtr 3854

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855

This theorem is referenced by:  trel3  3862  trintssm  3870  ordtr1  4125  suctr  4158  trsuc  4159  ordn2lp  4269