Theorem suctr 4476

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 110	. . . . 5 |- ( ( z e. y /\ y e. suc A ) -> y e. suc A )
2		vex 2776	. . . . . 6 |- y e. _V
3	2	elsuc 4461	. . . . 5 |- ( y e. suc A <-> ( y e. A \/ y = A ) )
4	1, 3	sylib 122	. . . 4 |- ( ( z e. y /\ y e. suc A ) -> ( y e. A \/ y = A ) )
5		simpl 109	. . . . . . 7 |- ( ( z e. y /\ y e. suc A ) -> z e. y )
6		eleq2 2270	. . . . . . 7 |- ( y = A -> ( z e. y <-> z e. A ) )
7	5, 6	syl5ibcom 155	. . . . . 6 |- ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. A ) )
8		elelsuc 4464	. . . . . 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 4157	. . . . . . . . 9 |- ( Tr A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
11	10	expd 258	. . . . . . . 8 |- ( Tr A -> ( z e. y -> ( y e. A -> z e. A ) ) )
12	11	adantrd 279	. . . . . . 7 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. A ) ) )
13	12, 8	syl8 71	. . . . . 6 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. suc A ) ) )
14		jao 757	. . . . . 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 43	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( ( y e. A \/ y = A ) -> z e. suc A ) ) )
17	4, 16	mpdi 43	. . 3 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
18	17	alrimivv 1899	. 2 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
19		dftr2 4152	. 2 |- ( Tr suc A <-> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
20	18, 19	sylibr 134	1 |- ( Tr A -> Tr suc A )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   A.wal 1371   = wceq 1373   e. wcel 2177   Tr wtr 4150   suc csuc 4420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-uni 3857  df-tr 4151  df-suc 4426

This theorem is referenced by:  ordsucim  4556  ordom  4663