Theorem ordsucim 4533

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 4410	. . 3 |- ( Ord A -> Tr A )
2		suctr 4453	. . 3 |- ( Tr A -> Tr suc A )
3	1, 2	syl 14	. 2 |- ( Ord A -> Tr suc A )
4		df-suc 4403	. . . . . 6 |- suc A = ( A u. { A } )
5	4	eleq2i 2260	. . . . 5 |- ( x e. suc A <-> x e. ( A u. { A } ) )
6		elun 3301	. . . . 5 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
7		velsn 3636	. . . . . 6 |- ( x e. { A } <-> x = A )
8	7	orbi2i 763	. . . . 5 |- ( ( x e. A \/ x e. { A } ) <-> ( x e. A \/ x = A ) )
9	5, 6, 8	3bitri 206	. . . 4 |- ( x e. suc A <-> ( x e. A \/ x = A ) )
10		dford3 4399	. . . . . . . 8 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
11	10	simprbi 275	. . . . . . 7 |- ( Ord A -> A. x e. A Tr x )
12		df-ral 2477	. . . . . . 7 |- ( A. x e. A Tr x <-> A. x ( x e. A -> Tr x ) )
13	11, 12	sylib 122	. . . . . 6 |- ( Ord A -> A. x ( x e. A -> Tr x ) )
14	13	19.21bi 1569	. . . . 5 |- ( Ord A -> ( x e. A -> Tr x ) )
15		treq 4134	. . . . . 6 |- ( x = A -> ( Tr x <-> Tr A ) )
16	1, 15	syl5ibrcom 157	. . . . 5 |- ( Ord A -> ( x = A -> Tr x ) )
17	14, 16	jaod 718	. . . 4 |- ( Ord A -> ( ( x e. A \/ x = A ) -> Tr x ) )
18	9, 17	biimtrid 152	. . 3 |- ( Ord A -> ( x e. suc A -> Tr x ) )
19	18	ralrimiv 2566	. 2 |- ( Ord A -> A. x e. suc A Tr x )
20		dford3 4399	. 2 |- ( Ord suc A <-> ( Tr suc A /\ A. x e. suc A Tr x ) )
21	3, 19, 20	sylanbrc 417	1 |- ( Ord A -> Ord suc A )

Syntax hints:   -> wi 4   \/ wo 709   A.wal 1362   = wceq 1364   e. wcel 2164   A.wral 2472   u. cun 3152   {csn 3619   Tr wtr 4128   Ord word 4394   suc csuc 4397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-uni 3837  df-tr 4129  df-iord 4398  df-suc 4403

This theorem is referenced by:  onsuc  4534  ordsucg  4535  onsucsssucr  4542  ordtriexmidlem  4552  2ordpr  4557  ordsuc  4596  nnsucsssuc  6547