Theorem suctrOLD 5809

Description: Obsolete proof of suctr 5808 as of 24-Sep-2021. (Contributed by Alan Sare, 11-Apr-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem suctrOLD

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

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 |- ( ( z e. y /\ y e. suc A ) -> y e. suc A )
2		vex 3203	. . . . . 6 |- y e. _V
3	2	elsuc 5794	. . . . 5 |- ( y e. suc A <-> ( y e. A \/ y = A ) )
4	1, 3	sylib 208	. . . 4 |- ( ( z e. y /\ y e. suc A ) -> ( y e. A \/ y = A ) )
5		simpl 473	. . . . . . 7 |- ( ( z e. y /\ y e. suc A ) -> z e. y )
6		eleq2 2690	. . . . . . 7 |- ( y = A -> ( z e. y <-> z e. A ) )
7	5, 6	syl5ibcom 235	. . . . . 6 |- ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. A ) )
8		elelsuc 5797	. . . . . 6 |- ( z e. A -> z e. suc A )
9	7, 8	syl6 35	. . . . 5 |- ( ( z e. y /\ y e. suc A ) -> ( y = A -> z e. suc A ) )
10		trel 4759	. . . . . . . . 9 |- ( Tr A -> ( ( z e. y /\ y e. A ) -> z e. A ) )
11	10	expd 452	. . . . . . . 8 |- ( Tr A -> ( z e. y -> ( y e. A -> z e. A ) ) )
12	11	adantrd 484	. . . . . . 7 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. A ) ) )
13	12, 8	syl8 76	. . . . . 6 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( y e. A -> z e. suc A ) ) )
14		jao 534	. . . . . 6 |- ( ( y e. A -> z e. suc A ) -> ( ( y = A -> z e. suc A ) -> ( ( y e. A \/ y = A ) -> z e. suc A ) ) )
15	13, 14	syl6 35	. . . . 5 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( ( y = A -> z e. suc A ) -> ( ( y e. A \/ y = A ) -> z e. suc A ) ) ) )
16	9, 15	mpdi 45	. . . 4 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> ( ( y e. A \/ y = A ) -> z e. suc A ) ) )
17	4, 16	mpdi 45	. . 3 |- ( Tr A -> ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
18	17	alrimivv 1856	. 2 |- ( Tr A -> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
19		dftr2 4754	. 2 |- ( Tr suc A <-> A. z A. y ( ( z e. y /\ y e. suc A ) -> z e. suc A ) )
20	18, 19	sylibr 224	1 |- ( Tr A -> Tr suc A )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   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-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)