Theorem ordsucim 4307

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 4196	. . 3 |- ( Ord A -> Tr A )
2		suctr 4239	. . 3 |- ( Tr A -> Tr suc A )
3	1, 2	syl 14	. 2 |- ( Ord A -> Tr suc A )
4		df-suc 4189	. . . . . 6 |- suc A = ( A u. { A } )
5	4	eleq2i 2154	. . . . 5 |- ( x e. suc A <-> x e. ( A u. { A } ) )
6		elun 3139	. . . . 5 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
7		velsn 3458	. . . . . 6 |- ( x e. { A } <-> x = A )
8	7	orbi2i 714	. . . . 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 4185	. . . . . . . 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 2364	. . . . . . 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 1495	. . . . 5 |- ( Ord A -> ( x e. A -> Tr x ) )
15		treq 3934	. . . . . 6 |- ( x = A -> ( Tr x <-> Tr A ) )
16	1, 15	syl5ibrcom 155	. . . . 5 |- ( Ord A -> ( x = A -> Tr x ) )
17	14, 16	jaod 672	. . . 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 2445	. 2 |- ( Ord A -> A. x e. suc A Tr x )
20		dford3 4185	. 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 664   A.wal 1287   = wceq 1289   e. wcel 1438   A.wral 2359   u. cun 2995   {csn 3441   Tr wtr 3928   Ord word 4180   suc csuc 4183

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-uni 3649  df-tr 3929  df-iord 4184  df-suc 4189

This theorem is referenced by:  suceloni  4308  ordsucg  4309  onsucsssucr  4316  ordtriexmidlem  4326  2ordpr  4330  ordsuc  4369  nnsucsssuc  6235