Theorem suctr 4176

Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)

Assertion

Ref	Expression
suctr	|- ( Tr A -> Tr suc A )

Proof of Theorem suctr

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

Step	Hyp	Ref	Expression
1		simpr 108	. . . . 5 |- ( ( z e. y /\ y e. suc A ) -> y e. suc A )
2		vex 2604	. . . . . 6 |- y e. _V
3	2	elsuc 4161	. . . . 5 |- ( y e. suc A <-> ( y e. A \/ y = A ) )
4	1, 3	sylib 120	. . . 4 |- ( ( z e. y /\ y e. suc A ) -> ( y e. A \/ y = A ) )
5		simpl 107	. . . . . . 7 |- ( ( z e. y /\ y e. suc A ) -> z e. y )
6		eleq2 2142	. . . . . . 7 |- ( y = A -> ( z e. y <-> z e. A ) )
7	5, 6	syl5ibcom 153	. . . . . 6 |- ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. A ) )
8		elelsuc 4164	. . . . . 6 |- ( z e. A -> z e. suc A )
9	7, 8	syl6 33	. . . . 5 |- ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. suc A ) )
10		trel 3882	. . . . . . . . 9 |- ( Tr A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
11	10	expd 254	. . . . . . . 8 |- ( Tr A -> ( z e. y -> ( y e. A -> z e. A ) ) )
12	11	adantrd 273	. . . . . . 7 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. A ) ) )
13	12, 8	syl8 70	. . . . . 6 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. suc A ) ) )
14		jao 704	. . . . . 6 |- ( ( y e. A -> z e. suc A ) -> ( ( y = A -> z e. suc A ) -> ( ( y e. A \/ y = A ) -> z e. suc A ) ) )
15	13, 14	syl6 33	. . . . 5 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( ( y = A -> z e. suc A ) -> ( ( y e. A \/ y = A ) -> z e. suc A ) ) ) )
16	9, 15	mpdi 42	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( ( y e. A \/ y = A ) -> z e. suc A ) ) )
17	4, 16	mpdi 42	. . 3 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
18	17	alrimivv 1796	. 2 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
19		dftr2 3877	. 2 |- ( Tr suc A <-> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
20	18, 19	sylibr 132	1 |- ( Tr A -> Tr suc A )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   A.wal 1282   = wceq 1284   e. wcel 1433   Tr wtr 3875   suc csuc 4120

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-uni 3602  df-tr 3876  df-suc 4126

This theorem is referenced by:  ordsucim  4244  ordom  4347