Theorem ordsucim 4493

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 4372	. . 3 |- ( Ord A -> Tr A )
2		suctr 4415	. . 3 |- ( Tr A -> Tr suc A )
3	1, 2	syl 14	. 2 |- ( Ord A -> Tr suc A )
4		df-suc 4365	. . . . . 6 |- suc A = ( A u. { A } )
5	4	eleq2i 2242	. . . . 5 |- ( x e. suc A <-> x e. ( A u. { A } ) )
6		elun 3274	. . . . 5 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
7		velsn 3606	. . . . . 6 |- ( x e. { A } <-> x = A )
8	7	orbi2i 762	. . . . 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 4361	. . . . . . . 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 2458	. . . . . . 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 1556	. . . . 5 |- ( Ord A -> ( x e. A -> Tr x ) )
15		treq 4102	. . . . . 6 |- ( x = A -> ( Tr x <-> Tr A ) )
16	1, 15	syl5ibrcom 157	. . . . 5 |- ( Ord A -> ( x = A -> Tr x ) )
17	14, 16	jaod 717	. . . 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 2547	. 2 |- ( Ord A -> A. x e. suc A Tr x )
20		dford3 4361	. 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 708   A.wal 1351   = wceq 1353   e. wcel 2146   A.wral 2453   u. cun 3125   {csn 3589   Tr wtr 4096   Ord word 4356   suc csuc 4359

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-sn 3595  df-uni 3806  df-tr 4097  df-iord 4360  df-suc 4365

This theorem is referenced by:  suceloni  4494  ordsucg  4495  onsucsssucr  4502  ordtriexmidlem  4512  2ordpr  4517  ordsuc  4556  nnsucsssuc  6483