Theorem ordsucim 4598

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 4475	. . 3 |- ( Ord A -> Tr A )
2		suctr 4518	. . 3 |- ( Tr A -> Tr suc A )
3	1, 2	syl 14	. 2 |- ( Ord A -> Tr suc A )
4		df-suc 4468	. . . . . 6 |- suc A = ( A u. { A } )
5	4	eleq2i 2298	. . . . 5 |- ( x e. suc A <-> x e. ( A u. { A } ) )
6		elun 3348	. . . . 5 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
7		velsn 3686	. . . . . 6 |- ( x e. { A } <-> x = A )
8	7	orbi2i 769	. . . . 5 |- ( ( x e. A \/ x e. { A } ) <-> ( x e. A \/ x = A ) )
9	5, 6, 8	3bitri 206	. . . 4 |- ( x e. suc A <-> ( x e. A \/ x = A ) )
10		dford3 4464	. . . . . . . 8 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
11	10	simprbi 275	. . . . . . 7 |- ( Ord A -> A. x e. A Tr x )
12		df-ral 2515	. . . . . . 7 |- ( A. x e. A Tr x <-> A. x ( x e. A -> Tr x ) )
13	11, 12	sylib 122	. . . . . 6 |- ( Ord A -> A. x ( x e. A -> Tr x ) )
14	13	19.21bi 1606	. . . . 5 |- ( Ord A -> ( x e. A -> Tr x ) )
15		treq 4193	. . . . . 6 |- ( x = A -> ( Tr x <-> Tr A ) )
16	1, 15	syl5ibrcom 157	. . . . 5 |- ( Ord A -> ( x = A -> Tr x ) )
17	14, 16	jaod 724	. . . 4 |- ( Ord A -> ( ( x e. A \/ x = A ) -> Tr x ) )
18	9, 17	biimtrid 152	. . 3 |- ( Ord A -> ( x e. suc A -> Tr x ) )
19	18	ralrimiv 2604	. 2 |- ( Ord A -> A. x e. suc A Tr x )
20		dford3 4464	. 2 |- ( Ord suc A <-> ( Tr suc A /\ A. x e. suc A Tr x ) )
21	3, 19, 20	sylanbrc 417	1 |- ( Ord A -> Ord suc A )

Syntax hints:   -> wi 4   \/ wo 715   A.wal 1395   = wceq 1397   e. wcel 2202   A.wral 2510   u. cun 3198   {csn 3669   Tr wtr 4187   Ord word 4459   suc csuc 4462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-uni 3894  df-tr 4188  df-iord 4463  df-suc 4468

This theorem is referenced by:  onsuc  4599  ordsucg  4600  onsucsssucr  4607  ordtriexmidlem  4617  2ordpr  4622  ordsuc  4661  nnsucsssuc  6659