Theorem trel 3941

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 3936	. 2 |- ( Tr A <-> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
2		eleq12 2152	. . . . . 6 |- ( ( y = B /\ x = C ) -> ( y e. x <-> B e. C ) )
3		eleq1 2150	. . . . . . 7 |- ( x = C -> ( x e. A <-> C e. A ) )
4	3	adantl 271	. . . . . 6 |- ( ( y = B /\ x = C ) -> ( x e. A <-> C e. A ) )
5	2, 4	anbi12d 457	. . . . 5 |- ( ( y = B /\ x = C ) -> ( ( y e. x /\ x e. A ) <-> ( B e. C /\ C e. A ) ) )
6		eleq1 2150	. . . . . 6 |- ( y = B -> ( y e. A <-> B e. A ) )
7	6	adantr 270	. . . . 5 |- ( ( y = B /\ x = C ) -> ( y e. A <-> B e. A ) )
8	5, 7	imbi12d 232	. . . 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 2709	. . 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 51	. 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 119	1 |- ( Tr A -> ( ( B e. C /\ C e. A ) -> B e. A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1287   = wceq 1289   e. wcel 1438   Tr wtr 3934

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-uni 3652  df-tr 3935

This theorem is referenced by:  trel3  3942  trintssmOLD  3951  ordtr1  4213  suctr  4246  trsuc  4247  ordn2lp  4359