Theorem ordsucim 4226

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 4115	. . 3 |- ( Ord A -> Tr A )
2		suctr 4158	. . 3 |- ( Tr A -> Tr suc A )
3	1, 2	syl 14	. 2 |- ( Ord A -> Tr suc A )
4		df-suc 4108	. . . . . 6 |- suc A = ( A u. { A } )
5	4	eleq2i 2104	. . . . 5 |- ( x e. suc A <-> x e. ( A u. { A } ) )
6		elun 3084	. . . . 5 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
7		velsn 3392	. . . . . 6 |- ( x e. { A } <-> x = A )
8	7	orbi2i 679	. . . . 5 |- ( ( x e. A \/ x e. { A } ) <-> ( x e. A \/ x = A ) )
9	5, 6, 8	3bitri 195	. . . 4 |- ( x e. suc A <-> ( x e. A \/ x = A ) )
10		dford3 4104	. . . . . . . 8 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
11	10	simprbi 260	. . . . . . 7 |- ( Ord A -> A. x e. A Tr x )
12		df-ral 2311	. . . . . . 7 |- ( A. x e. A Tr x <-> A. x ( x e. A -> Tr x ) )
13	11, 12	sylib 127	. . . . . 6 |- ( Ord A -> A. x ( x e. A -> Tr x ) )
14	13	19.21bi 1450	. . . . 5 |- ( Ord A -> ( x e. A -> Tr x ) )
15		treq 3860	. . . . . 6 |- ( x = A -> ( Tr x <-> Tr A ) )
16	1, 15	syl5ibrcom 146	. . . . 5 |- ( Ord A -> ( x = A -> Tr x ) )
17	14, 16	jaod 637	. . . 4 |- ( Ord A -> ( ( x e. A \/ x = A ) -> Tr x ) )
18	9, 17	syl5bi 141	. . 3 |- ( Ord A -> ( x e. suc A -> Tr x ) )
19	18	ralrimiv 2391	. 2 |- ( Ord A -> A. x e. suc A Tr x )
20		dford3 4104	. 2 |- ( Ord suc A <-> ( Tr suc A /\ A. x e. suc A Tr x ) )
21	3, 19, 20	sylanbrc 394	1 |- ( Ord A -> Ord suc A )

Syntax hints:   -> wi 4   \/ wo 629   A.wal 1241   = wceq 1243   e. wcel 1393   A.wral 2306   u. cun 2915   {csn 3375   Tr wtr 3854   Ord word 4099   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103  df-suc 4108

This theorem is referenced by:  suceloni  4227  ordsucg  4228  onsucsssucr  4235  ordtriexmidlem  4245  2ordpr  4249  ordsuc  4287  nnsucsssuc  6071