Theorem suctrALT 39061

Description: The successor of a transitive class is transitive. The proof of http://us.metamath.org/other/completeusersproof/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctrALT 39061 using completeusersproof, which is verified by the Metamath program. The proof of http://us.metamath.org/other/completeusersproof/suctrro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. See suctr 5808 for the original proof. (Contributed by Alan Sare, 11-Apr-2009.) (Revised by Alan Sare, 12-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem suctrALT

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

Step	Hyp	Ref	Expression
1		sssucid 5802	. . . . . . 7 |- A C_ suc A
2		id 22	. . . . . . . 8 |- ( Tr A -> Tr A )
3		id 22	. . . . . . . . 9 |- ( ( z e. y /\ y e. suc A ) -> ( z e. y /\ y e. suc A ) )
4	3	simpld 475	. . . . . . . 8 |- ( ( z e. y /\ y e. suc A ) -> z e. y )
5		id 22	. . . . . . . 8 |- ( y e. A -> y e. A )
6		trel 4759	. . . . . . . . . 10 |- ( Tr A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
7	6	3impib 1262	. . . . . . . . 9 |- ( ( Tr A /\ z e. y /\ y e. A ) -> z e. A )
8	7	idiALT 38683	. . . . . . . 8 |- ( ( Tr A /\ z e. y /\ y e. A ) -> z e. A )
9	2, 4, 5, 8	syl3an 1368	. . . . . . 7 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) /\ y e. A ) -> z e. A )
10	1, 9	sseldi 3601	. . . . . 6 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) /\ y e. A ) -> z e. suc A )
11	10	3expia 1267	. . . . 5 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) ) -> ( y e. A -> z e. suc A ) )
12	4	adantr 481	. . . . . . . . 9 |- ( ( ( z e. y /\ y e. suc A ) /\ y = A ) -> z e. y )
13		id 22	. . . . . . . . . 10 |- ( y = A -> y = A )
14	13	adantl 482	. . . . . . . . 9 |- ( ( ( z e. y /\ y e. suc A ) /\ y = A ) -> y = A )
15	12, 14	eleqtrd 2703	. . . . . . . 8 |- ( ( ( z e. y /\ y e. suc A ) /\ y = A ) -> z e. A )
16	1, 15	sseldi 3601	. . . . . . 7 |- ( ( ( z e. y /\ y e. suc A ) /\ y = A ) -> z e. suc A )
17	16	ex 450	. . . . . 6 |- ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. suc A ) )
18	17	adantl 482	. . . . 5 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) ) -> ( y = A -> z e. suc A ) )
19	3	simprd 479	. . . . . . 7 |- ( ( z e. y /\ y e. suc A ) -> y e. suc A )
20		elsuci 5791	. . . . . . 7 |- ( y e. suc A -> ( y e. A \/ y = A ) )
21	19, 20	syl 17	. . . . . 6 |- ( ( z e. y /\ y e. suc A ) -> ( y e. A \/ y = A ) )
22	21	adantl 482	. . . . 5 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) ) -> ( y e. A \/ y = A ) )
23	11, 18, 22	mpjaod 396	. . . 4 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) ) -> z e. suc A )
24	23	ex 450	. . 3 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
25	24	alrimivv 1856	. 2 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
26		dftr2 4754	. . 3 |- ( Tr suc A <-> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
27	26	biimpri 218	. 2 |- ( A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) -> Tr suc A )
28	25, 27	syl 17	1 |- ( Tr A -> Tr suc A )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   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-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)