Theorem suctrALT2 31886

Description: Virtual deduction proof of suctr 4905. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 31885 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 4899	. . . . 5 |- A C_ suc A
2		trel 4495	. . . . . . 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 3453	. . . . 5 |- ( A C_ suc A -> ( z e. A -> z e. suc A ) )
6	1, 4, 5	ee03 31787	. . . 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 2525	. . . . . . 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 31787	. . . 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 4888	. . . . 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 31519	. . 3 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
19	18	alrimivv 1687	. 2 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
20		dftr2 4490	. 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 1368   = wceq 1370   e. wcel 1758   C_ wss 3431   Tr wtr 4488   suc csuc 4824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-v 3074  df-un 3436  df-in 3438  df-ss 3445  df-sn 3981  df-uni 4195  df-tr 4489  df-suc 4828

This theorem is referenced by: (None)