Theorem suctrALT2 33780

Description: Virtual deduction proof of suctr 4970. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 33779 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 4964	. . . . 5 |- A C_ suc A
2		trel 4557	. . . . . . 7 |- ( Tr A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
3	2	expd 436	. . . . . 6 |- ( Tr A -> ( z e. y -> ( y e. A -> z e. A ) ) )
4	3	adantrd 468	. . . . 5 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. A ) ) )
5		ssel 3493	. . . . 5 |- ( A C_ suc A -> ( z e. A -> z e. suc A ) )
6	1, 4, 5	ee03 33681	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. suc A ) ) )
7		simpl 457	. . . . . . 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 2530	. . . . . . 7 |- ( y = A -> ( z e. y <-> z e. A ) )
10	9	biimpcd 224	. . . . . 6 |- ( z e. y -> ( y = A -> z e. A ) )
11	8, 10	syl6 33	. . . . 5 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. A ) ) )
12	1, 11, 5	ee03 33681	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. suc A ) ) )
13		simpr 461	. . . . . 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 4953	. . . . 5 |- ( y e. suc A -> ( y e. A \/ y = A ) )
16	14, 15	syl6 33	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A \/ y = A ) ) )
17		jao 512	. . . 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 33414	. . 3 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
19	18	alrimivv 1721	. 2 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
20		dftr2 4552	. 2 |- ( Tr suc A <-> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
21	19, 20	sylibr 212	1 |- ( Tr A -> Tr suc A )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   A.wal 1393   = wceq 1395   e. wcel 1819   C_ wss 3471   Tr wtr 4550   suc csuc 4889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3476  df-in 3478  df-ss 3485  df-sn 4033  df-uni 4252  df-tr 4551  df-suc 4893

This theorem is referenced by: (None)