Theorem suctrALTcf 39158

Description: The sucessor of a transitive class is transitive. suctrALTcf 39158, using conventional notation, was translated from virtual deduction form, suctrALTcfVD 39159, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem suctrALTcf

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

Step	Hyp	Ref	Expression
1		sssucid 5802	. . . . . . . 8 |- A C_ suc A
2		id 22	. . . . . . . . 9 |- ( Tr A -> Tr A )
3		id 22	. . . . . . . . . 10 |- ( ( z e. y /\ y e. suc A ) -> ( z e. y /\ y e. suc A ) )
4		simpl 473	. . . . . . . . . 10 |- ( ( z e. y /\ y e. suc A ) -> z e. y )
5	3, 4	syl 17	. . . . . . . . 9 |- ( ( z e. y /\ y e. suc A ) -> z e. y )
6		id 22	. . . . . . . . 9 |- ( y e. A -> y e. A )
7		trel 4759	. . . . . . . . . 10 |- ( Tr A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
8	7	3impib 1262	. . . . . . . . 9 |- ( ( Tr A /\ z e. y /\ y e. A ) -> z e. A )
9	2, 5, 6, 8	syl3an 1368	. . . . . . . 8 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) /\ y e. A ) -> z e. A )
10		ssel2 3598	. . . . . . . 8 |- ( ( A C_ suc A /\ z e. A ) -> z e. suc A )
11	1, 9, 10	eel0321old 38941	. . . . . . 7 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) /\ y e. A ) -> z e. suc A )
12	11	3expia 1267	. . . . . 6 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) ) -> ( y e. A -> z e. suc A ) )
13		id 22	. . . . . . . . 9 |- ( y = A -> y = A )
14		eleq2 2690	. . . . . . . . . 10 |- ( y = A -> ( z e. y <-> z e. A ) )
15	14	biimpac 503	. . . . . . . . 9 |- ( ( z e. y /\ y = A ) -> z e. A )
16	5, 13, 15	syl2an 494	. . . . . . . 8 |- ( ( ( z e. y /\ y e. suc A ) /\ y = A ) -> z e. A )
17	1, 16, 10	eel021old 38925	. . . . . . 7 |- ( ( ( z e. y /\ y e. suc A ) /\ y = A ) -> z e. suc A )
18	17	ex 450	. . . . . 6 |- ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. suc A ) )
19		simpr 477	. . . . . . . 8 |- ( ( z e. y /\ y e. suc A ) -> y e. suc A )
20	3, 19	syl 17	. . . . . . 7 |- ( ( z e. y /\ y e. suc A ) -> y e. suc A )
21		elsuci 5791	. . . . . . 7 |- ( y e. suc A -> ( y e. A \/ y = A ) )
22	20, 21	syl 17	. . . . . 6 |- ( ( z e. y /\ y e. suc A ) -> ( y e. A \/ y = A ) )
23		jao 534	. . . . . . 7 |- ( ( y e. A -> z e. suc A ) -> ( ( y = A -> z e. suc A ) -> ( ( y e. A \/ y = A ) -> z e. suc A ) ) )
24	23	3imp 1256	. . . . . 6 |- ( ( ( y e. A -> z e. suc A ) /\ ( y = A -> z e. suc A ) /\ ( y e. A \/ y = A ) ) -> z e. suc A )
25	12, 18, 22, 24	eel2122old 38943	. . . . 5 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) ) -> z e. suc A )
26	25	ex 450	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
27	26	alrimivv 1856	. . 3 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
28		dftr2 4754	. . . 4 |- ( Tr suc A <-> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
29	28	biimpri 218	. . 3 |- ( A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) -> Tr suc A )
30	27, 29	syl 17	. 2 |- ( Tr A -> Tr suc A )
31	30	iin1 38788	1 |- ( Tr A -> Tr suc A )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   C_ wss 3574   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-3an 1039  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: (None)