Theorem ordsucim 4548

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 4425	. . 3 |- ( Ord A -> Tr A )
2		suctr 4468	. . 3 |- ( Tr A -> Tr suc A )
3	1, 2	syl 14	. 2 |- ( Ord A -> Tr suc A )
4		df-suc 4418	. . . . . 6 |- suc A = ( A u. { A } )
5	4	eleq2i 2272	. . . . 5 |- ( x e. suc A <-> x e. ( A u. { A } ) )
6		elun 3314	. . . . 5 |- ( x e. ( A u. { A } ) <-> ( x e. A \/ x e. { A } ) )
7		velsn 3650	. . . . . 6 |- ( x e. { A } <-> x = A )
8	7	orbi2i 764	. . . . 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 4414	. . . . . . . 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 2489	. . . . . . 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 1581	. . . . 5 |- ( Ord A -> ( x e. A -> Tr x ) )
15		treq 4148	. . . . . 6 |- ( x = A -> ( Tr x <-> Tr A ) )
16	1, 15	syl5ibrcom 157	. . . . 5 |- ( Ord A -> ( x = A -> Tr x ) )
17	14, 16	jaod 719	. . . 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 2578	. 2 |- ( Ord A -> A. x e. suc A Tr x )
20		dford3 4414	. 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 710   A.wal 1371   = wceq 1373   e. wcel 2176   A.wral 2484   u. cun 3164   {csn 3633   Tr wtr 4142   Ord word 4409   suc csuc 4412

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-uni 3851  df-tr 4143  df-iord 4413  df-suc 4418

This theorem is referenced by:  onsuc  4549  ordsucg  4550  onsucsssucr  4557  ordtriexmidlem  4567  2ordpr  4572  ordsuc  4611  nnsucsssuc  6578