Theorem suctr 4184

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 2605	. . . . . 6 |- y e. _V
3	2	elsuc 4169	. . . . 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 2143	. . . . . . 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 4172	. . . . . 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 3890	. . . . . . . . 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 705	. . . . . 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 1797	. 2 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
19		dftr2 3885	. 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 662   A.wal 1283   = wceq 1285   e. wcel 1434   Tr wtr 3883   suc csuc 4128

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-uni 3610  df-tr 3884  df-suc 4134

This theorem is referenced by:  ordsucim  4252  ordom  4355