Theorem suctrALT2 39072

Description: Virtual deduction proof of suctr 5808. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 39071 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 5802	. . . . 5 |- A C_ suc A
2		trel 4759	. . . . . . 7 |- ( Tr A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
3	2	expd 452	. . . . . 6 |- ( Tr A -> ( z e. y -> ( y e. A -> z e. A ) ) )
4	3	adantrd 484	. . . . 5 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. A ) ) )
5		ssel 3597	. . . . 5 |- ( A C_ suc A -> ( z e. A -> z e. suc A ) )
6	1, 4, 5	ee03 38968	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. suc A ) ) )
7		simpl 473	. . . . . . 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 2690	. . . . . . 7 |- ( y = A -> ( z e. y <-> z e. A ) )
10	9	biimpcd 239	. . . . . 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 38968	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. suc A ) ) )
13		simpr 477	. . . . . 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 5791	. . . . 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 534	. . . 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 38708	. . 3 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
19	18	alrimivv 1856	. 2 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
20		dftr2 4754	. 2 |- ( Tr suc A <-> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
21	19, 20	sylibr 224	1 |- ( Tr A -> Tr suc A )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   C_ wss 3574   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-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)