Theorem ordsucim 4244

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 4133	. . 3 |- ( Ord A -> Tr A )
2		suctr 4176	. . 3 |- ( Tr A -> Tr suc A )
3	1, 2	syl 14	. 2 |- ( Ord A -> Tr suc A )
4		df-suc 4126	. . . . . 6 |- suc A = ( A u. { A } )
5	4	eleq2i 2145	. . . . 5 |- ( x e. suc A <-> x e. ( A u. { A } ) )
6		elun 3113	. . . . 5 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
7		velsn 3415	. . . . . 6 |- ( x e. { A } <-> x = A )
8	7	orbi2i 711	. . . . 5 |- ( ( x e. A \/ x e. { A } ) <-> ( x e. A \/ x = A ) )
9	5, 6, 8	3bitri 204	. . . 4 |- ( x e. suc A <-> ( x e. A \/ x = A ) )
10		dford3 4122	. . . . . . . 8 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
11	10	simprbi 269	. . . . . . 7 |- ( Ord A -> A. x e. A Tr x )
12		df-ral 2353	. . . . . . 7 |- ( A. x e. A Tr x <-> A. x ( x e. A -> Tr x ) )
13	11, 12	sylib 120	. . . . . 6 |- ( Ord A -> A. x ( x e. A -> Tr x ) )
14	13	19.21bi 1490	. . . . 5 |- ( Ord A -> ( x e. A -> Tr x ) )
15		treq 3881	. . . . . 6 |- ( x = A -> ( Tr x <-> Tr A ) )
16	1, 15	syl5ibrcom 155	. . . . 5 |- ( Ord A -> ( x = A -> Tr x ) )
17	14, 16	jaod 669	. . . 4 |- ( Ord A -> ( ( x e. A \/ x = A ) -> Tr x ) )
18	9, 17	syl5bi 150	. . 3 |- ( Ord A -> ( x e. suc A -> Tr x ) )
19	18	ralrimiv 2433	. 2 |- ( Ord A -> A. x e. suc A Tr x )
20		dford3 4122	. 2 |- ( Ord suc A <-> ( Tr suc A /\ A. x e. suc A Tr x ) )
21	3, 19, 20	sylanbrc 408	1 |- ( Ord A -> Ord suc A )

Syntax hints:   -> wi 4   \/ wo 661   A.wal 1282   = wceq 1284   e. wcel 1433   A.wral 2348   u. cun 2971   {csn 3398   Tr wtr 3875   Ord word 4117   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-uni 3602  df-tr 3876  df-iord 4121  df-suc 4126

This theorem is referenced by:  suceloni  4245  ordsucg  4246  onsucsssucr  4253  ordtriexmidlem  4263  2ordpr  4267  ordsuc  4306  nnsucsssuc  6094