Theorem suctr 4158

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 103	. . . . 5 |- ( ( z e. y /\ y e. suc A ) -> y e. suc A )
2		vex 2560	. . . . . 6 |- y e. _V
3	2	elsuc 4143	. . . . 5 |- ( y e. suc A <-> ( y e. A \/ y = A ) )
4	1, 3	sylib 127	. . . 4 |- ( ( z e. y /\ y e. suc A ) -> ( y e. A \/ y = A ) )
5		simpl 102	. . . . . . 7 |- ( ( z e. y /\ y e. suc A ) -> z e. y )
6		eleq2 2101	. . . . . . 7 |- ( y = A -> ( z e. y <-> z e. A ) )
7	5, 6	syl5ibcom 144	. . . . . 6 |- ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. A ) )
8		elelsuc 4146	. . . . . 6 |- ( z e. A -> z e. suc A )
9	7, 8	syl6 29	. . . . 5 |- ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. suc A ) )
10		trel 3861	. . . . . . . . 9 |- ( Tr A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
11	10	expd 245	. . . . . . . 8 |- ( Tr A -> ( z e. y -> ( y e. A -> z e. A ) ) )
12	11	adantrd 264	. . . . . . 7 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. A ) ) )
13	12, 8	syl8 65	. . . . . 6 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. suc A ) ) )
14		jao 672	. . . . . 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 29	. . . . 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 38	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( ( y e. A \/ y = A ) -> z e. suc A ) ) )
17	4, 16	mpdi 38	. . 3 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
18	17	alrimivv 1755	. 2 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
19		dftr2 3856	. 2 |- ( Tr suc A <-> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
20	18, 19	sylibr 137	1 |- ( Tr A -> Tr suc A )

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 629   A.wal 1241   = wceq 1243   e. wcel 1393   Tr wtr 3854   suc csuc 4102

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-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-uni 3581  df-tr 3855  df-suc 4108

This theorem is referenced by:  ordsucim  4226  ordom  4329