Theorem ordsucim 4424

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 4308	. . 3 |- ( Ord A -> Tr A )
2		suctr 4351	. . 3 |- ( Tr A -> Tr suc A )
3	1, 2	syl 14	. 2 |- ( Ord A -> Tr suc A )
4		df-suc 4301	. . . . . 6 |- suc A = ( A u. { A } )
5	4	eleq2i 2207	. . . . 5 |- ( x e. suc A <-> x e. ( A u. { A } ) )
6		elun 3222	. . . . 5 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
7		velsn 3549	. . . . . 6 |- ( x e. { A } <-> x = A )
8	7	orbi2i 752	. . . . 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 4297	. . . . . . . 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 2422	. . . . . . 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 1538	. . . . 5 |- ( Ord A -> ( x e. A -> Tr x ) )
15		treq 4040	. . . . . 6 |- ( x = A -> ( Tr x <-> Tr A ) )
16	1, 15	syl5ibrcom 156	. . . . 5 |- ( Ord A -> ( x = A -> Tr x ) )
17	14, 16	jaod 707	. . . 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 2507	. 2 |- ( Ord A -> A. x e. suc A Tr x )
20		dford3 4297	. 2 |- ( Ord suc A <-> ( Tr suc A /\ A. x e. suc A Tr x ) )
21	3, 19, 20	sylanbrc 414	1 |- ( Ord A -> Ord suc A )

Syntax hints:   -> wi 4   \/ wo 698   A.wal 1330   = wceq 1332   e. wcel 1481   A.wral 2417   u. cun 3074   {csn 3532   Tr wtr 4034   Ord word 4292   suc csuc 4295

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-uni 3745  df-tr 4035  df-iord 4296  df-suc 4301

This theorem is referenced by:  suceloni  4425  ordsucg  4426  onsucsssucr  4433  ordtriexmidlem  4443  2ordpr  4447  ordsuc  4486  nnsucsssuc  6396