Theorem suctr 5808

Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.)

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		elsuci 5791	. . . . . 6 |- ( y e. suc A -> ( y e. A \/ y = A ) )
2		trel 4759	. . . . . . . 8 |- ( Tr A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
3	2	expdimp 453	. . . . . . 7 |- ( ( Tr A /\ z e. y ) -> ( y e. A -> z e. A ) )
4		eleq2 2690	. . . . . . . . 9 |- ( y = A -> ( z e. y <-> z e. A ) )
5	4	biimpcd 239	. . . . . . . 8 |- ( z e. y -> ( y = A -> z e. A ) )
6	5	adantl 482	. . . . . . 7 |- ( ( Tr A /\ z e. y ) -> ( y = A -> z e. A ) )
7	3, 6	jaod 395	. . . . . 6 |- ( ( Tr A /\ z e. y ) -> ( ( y e. A \/ y = A ) -> z e. A ) )
8	1, 7	syl5 34	. . . . 5 |- ( ( Tr A /\ z e. y ) -> ( y e. suc A -> z e. A ) )
9	8	expimpd 629	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. A ) )
10		elelsuc 5797	. . . 4 |- ( z e. A -> z e. suc A )
11	9, 10	syl6 35	. . 3 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
12	11	alrimivv 1856	. 2 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
13		dftr2 4754	. 2 |- ( Tr suc A <-> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
14	12, 13	sylibr 224	1 |- ( Tr A -> Tr suc A )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   Tr wtr 4752   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-uni 4437  df-tr 4753  df-suc 5729

This theorem is referenced by:  dfon2lem3  31690  dfon2lem7  31694  dford3lem2  37594