Theorem suctrALT2 37100

Description: Virtual deduction proof of suctr 5523. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 37099 using the tools command file translate_without_overwriting_minimize_excluding_duplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
suctrALT2	|- ( Tr A -> Tr suc A )

Proof of Theorem suctrALT2

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

Step	Hyp	Ref	Expression
1		sssucid 5517	. . . . 5 |- A C_ suc A
2		trel 4523	. . . . . . 7 |- ( Tr A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
3	2	expd 438	. . . . . 6 |- ( Tr A -> ( z e. y -> ( y e. A -> z e. A ) ) )
4	3	adantrd 470	. . . . 5 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. A ) ) )
5		ssel 3459	. . . . 5 |- ( A C_ suc A -> ( z e. A -> z e. suc A ) )
6	1, 4, 5	ee03 36995	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. suc A ) ) )
7		simpl 459	. . . . . . 7 |- ( ( z e. y /\ y e. suc A ) -> z e. y )
8	7	a1i 11	. . . . . 6 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. y ) )
9		eleq2 2496	. . . . . . 7 |- ( y = A -> ( z e. y <-> z e. A ) )
10	9	biimpcd 228	. . . . . 6 |- ( z e. y -> ( y = A -> z e. A ) )
11	8, 10	syl6 35	. . . . 5 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. A ) ) )
12	1, 11, 5	ee03 36995	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. suc A ) ) )
13		simpr 463	. . . . . 6 |- ( ( z e. y /\ y e. suc A ) -> y e. suc A )
14	13	a1i 11	. . . . 5 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> y e. suc A ) )
15		elsuci 5506	. . . . 5 |- ( y e. suc A -> ( y e. A \/ y = A ) )
16	14, 15	syl6 35	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A \/ y = A ) ) )
17		jao 515	. . . 4 |- ( ( y e. A -> z e. suc A ) -> ( ( y = A -> z e. suc A ) -> ( ( y e. A \/ y = A ) -> z e. suc A ) ) )
18	6, 12, 16, 17	ee222 36722	. . 3 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
19	18	alrimivv 1765	. 2 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
20		dftr2 4518	. 2 |- ( Tr suc A <-> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
21	19, 20	sylibr 216	1 |- ( Tr A -> Tr suc A )

Syntax hints:   -> wi 4   \/ wo 370   /\ wa 371   A.wal 1436   = wceq 1438   e. wcel 1869   C_ wss 3437   Tr wtr 4516   suc csuc 5442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-v 3084  df-un 3442  df-in 3444  df-ss 3451  df-sn 3998  df-uni 4218  df-tr 4517  df-suc 5446

This theorem is referenced by: (None)