Theorem ordsucim 4477

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 4356	. . 3 |- ( Ord A -> Tr A )
2		suctr 4399	. . 3 |- ( Tr A -> Tr suc A )
3	1, 2	syl 14	. 2 |- ( Ord A -> Tr suc A )
4		df-suc 4349	. . . . . 6 |- suc A = ( A u. { A } )
5	4	eleq2i 2233	. . . . 5 |- ( x e. suc A <-> x e. ( A u. { A } ) )
6		elun 3263	. . . . 5 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
7		velsn 3593	. . . . . 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 4345	. . . . . . . 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 2449	. . . . . . 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 1546	. . . . 5 |- ( Ord A -> ( x e. A -> Tr x ) )
15		treq 4086	. . . . . 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 2538	. 2 |- ( Ord A -> A. x e. suc A Tr x )
20		dford3 4345	. 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 1341   = wceq 1343   e. wcel 2136   A.wral 2444   u. cun 3114   {csn 3576   Tr wtr 4080   Ord word 4340   suc csuc 4343

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-uni 3790  df-tr 4081  df-iord 4344  df-suc 4349

This theorem is referenced by:  suceloni  4478  ordsucg  4479  onsucsssucr  4486  ordtriexmidlem  4496  2ordpr  4501  ordsuc  4540  nnsucsssuc  6460