Theorem trel 4069

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 4064	. 2 |- ( Tr A <-> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
2		eleq12 2222	. . . . . 6 |- ( ( y = B /\ x = C ) -> ( y e. x <-> B e. C ) )
3		eleq1 2220	. . . . . . 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 2220	. . . . . 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 2803	. . 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 1333   = wceq 1335   e. wcel 2128   Tr wtr 4062

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-uni 3773  df-tr 4063

This theorem is referenced by:  trel3  4070  ordtr1  4347  suctr  4380  trsuc  4381  ordn2lp  4502