Theorem ordsucim 4330

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 4214	. . 3 |- ( Ord A -> Tr A )
2		suctr 4257	. . 3 |- ( Tr A -> Tr suc A )
3	1, 2	syl 14	. 2 |- ( Ord A -> Tr suc A )
4		df-suc 4207	. . . . . 6 |- suc A = ( A u. { A } )
5	4	eleq2i 2155	. . . . 5 |- ( x e. suc A <-> x e. ( A u. { A } ) )
6		elun 3142	. . . . 5 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
7		velsn 3467	. . . . . 6 |- ( x e. { A } <-> x = A )
8	7	orbi2i 715	. . . . 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 4203	. . . . . . . 8 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
11	10	simprbi 270	. . . . . . 7 |- ( Ord A -> A. x e. A Tr x )
12		df-ral 2365	. . . . . . 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 1496	. . . . 5 |- ( Ord A -> ( x e. A -> Tr x ) )
15		treq 3948	. . . . . 6 |- ( x = A -> ( Tr x <-> Tr A ) )
16	1, 15	syl5ibrcom 156	. . . . 5 |- ( Ord A -> ( x = A -> Tr x ) )
17	14, 16	jaod 673	. . . 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 2446	. 2 |- ( Ord A -> A. x e. suc A Tr x )
20		dford3 4203	. 2 |- ( Ord suc A <-> ( Tr suc A /\ A. x e. suc A Tr x ) )
21	3, 19, 20	sylanbrc 409	1 |- ( Ord A -> Ord suc A )

Syntax hints:   -> wi 4   \/ wo 665   A.wal 1288   = wceq 1290   e. wcel 1439   A.wral 2360   u. cun 2998   {csn 3450   Tr wtr 3942   Ord word 4198   suc csuc 4201

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-uni 3660  df-tr 3943  df-iord 4202  df-suc 4207

This theorem is referenced by:  suceloni  4331  ordsucg  4332  onsucsssucr  4339  ordtriexmidlem  4349  2ordpr  4353  ordsuc  4392  nnsucsssuc  6267