Theorem ordsucim 4485

Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)

Assertion

Ref	Expression
ordsucim	|- ( Ord A -> Ord suc A )

Proof of Theorem ordsucim

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtr 4364	. . 3 |- ( Ord A -> Tr A )
2		suctr 4407	. . 3 |- ( Tr A -> Tr suc A )
3	1, 2	syl 14	. 2 |- ( Ord A -> Tr suc A )
4		df-suc 4357	. . . . . 6 |- suc A = ( A u. { A } )
5	4	eleq2i 2238	. . . . 5 |- ( x e. suc A <-> x e. ( A u. { A } ) )
6		elun 3269	. . . . 5 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
7		velsn 3601	. . . . . 6 |- ( x e. { A } <-> x = A )
8	7	orbi2i 758	. . . . 5 |- ( ( x e. A \/ x e. { A } ) <-> ( x e. A \/ x = A ) )
9	5, 6, 8	3bitri 205	. . . 4 |- ( x e. suc A <-> ( x e. A \/ x = A ) )
10		dford3 4353	. . . . . . . 8 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
11	10	simprbi 273	. . . . . . 7 |- ( Ord A -> A. x e. A Tr x )
12		df-ral 2454	. . . . . . 7 |- ( A. x e. A Tr x <-> A. x ( x e. A -> Tr x ) )
13	11, 12	sylib 121	. . . . . 6 |- ( Ord A -> A. x ( x e. A -> Tr x ) )
14	13	19.21bi 1552	. . . . 5 |- ( Ord A -> ( x e. A -> Tr x ) )
15		treq 4094	. . . . . 6 |- ( x = A -> ( Tr x <-> Tr A ) )
16	1, 15	syl5ibrcom 156	. . . . 5 |- ( Ord A -> ( x = A -> Tr x ) )
17	14, 16	jaod 713	. . . 4 |- ( Ord A -> ( ( x e. A \/ x = A ) -> Tr x ) )
18	9, 17	syl5bi 151	. . 3 |- ( Ord A -> ( x e. suc A -> Tr x ) )
19	18	ralrimiv 2543	. 2 |- ( Ord A -> A. x e. suc A Tr x )
20		dford3 4353	. 2 |- ( Ord suc A <-> ( Tr suc A /\ A. x e. suc A Tr x ) )
21	3, 19, 20	sylanbrc 415	1 |- ( Ord A -> Ord suc A )

Syntax hints:   -> wi 4   \/ wo 704   A.wal 1347   = wceq 1349   e. wcel 2142   A.wral 2449   u. cun 3120   {csn 3584   Tr wtr 4088   Ord word 4348   suc csuc 4351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-sn 3590  df-uni 3798  df-tr 4089  df-iord 4352  df-suc 4357

This theorem is referenced by:  suceloni  4486  ordsucg  4487  onsucsssucr  4494  ordtriexmidlem  4504  2ordpr  4509  ordsuc  4548  nnsucsssuc  6475