Theorem suctrALT3 39160

Description: The successor of a transitive class is transitive. suctrALT3 39160 is the completed proof in conventional notation of the Virtual Deduction proof http://us.metamath.org/other/completeusersproof/suctralt3vd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 38785 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction (e.g., the sub-theorem whose assertion is step 19 used jaoded 38782). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem (e.g., the sub-theorem whose assertion is step 24 used dftr2 4754) . (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
suctrALT3	|- ( Tr A -> Tr suc A )

Proof of Theorem suctrALT3

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

Step	Hyp	Ref	Expression
1		sssucid 5802	. . . . . . . . 9 |- A C_ suc A
2		id 22	. . . . . . . . . 10 |- ( Tr A -> Tr A )
3		id 22	. . . . . . . . . . 11 |- ( ( z e. y /\ y e. suc A ) -> ( z e. y /\ y e. suc A ) )
4	3	simpld 475	. . . . . . . . . 10 |- ( ( z e. y /\ y e. suc A ) -> z e. y )
5		id 22	. . . . . . . . . 10 |- ( y e. A -> y e. A )
6	2, 4, 5	trelded 38781	. . . . . . . . 9 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) /\ y e. A ) -> z e. A )
7	1, 6	sseldi 3601	. . . . . . . 8 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) /\ y e. A ) -> z e. suc A )
8	7	3expia 1267	. . . . . . 7 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) ) -> ( y e. A -> z e. suc A ) )
9		id 22	. . . . . . . . . 10 |- ( y = A -> y = A )
10		eleq2 2690	. . . . . . . . . . 11 |- ( y = A -> ( z e. y <-> z e. A ) )
11	10	biimpac 503	. . . . . . . . . 10 |- ( ( z e. y /\ y = A ) -> z e. A )
12	4, 9, 11	syl2an 494	. . . . . . . . 9 |- ( ( ( z e. y /\ y e. suc A ) /\ y = A ) -> z e. A )
13	1, 12	sseldi 3601	. . . . . . . 8 |- ( ( ( z e. y /\ y e. suc A ) /\ y = A ) -> z e. suc A )
14	13	ex 450	. . . . . . 7 |- ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. suc A ) )
15	3	simprd 479	. . . . . . . 8 |- ( ( z e. y /\ y e. suc A ) -> y e. suc A )
16		elsuci 5791	. . . . . . . 8 |- ( y e. suc A -> ( y e. A \/ y = A ) )
17	15, 16	syl 17	. . . . . . 7 |- ( ( z e. y /\ y e. suc A ) -> ( y e. A \/ y = A ) )
18	8, 14, 17	jaoded 38782	. . . . . 6 |- ( ( ( Tr A /\ ( z e. y /\ y e. suc A ) ) /\ ( z e. y /\ y e. suc A ) /\ ( z e. y /\ y e. suc A ) ) -> z e. suc A )
19	18	un2122 39017	. . . . 5 |- ( ( Tr A /\ ( z e. y /\ y e. suc A ) ) -> z e. suc A )
20	19	ex 450	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
21	20	alrimivv 1856	. . 3 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
22		dftr2 4754	. . . 4 |- ( Tr suc A <-> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
23	22	biimpri 218	. . 3 |- ( A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) -> Tr suc A )
24	21, 23	syl 17	. 2 |- ( Tr A -> Tr suc A )
25	24	idiALT 38683	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)