Theorem onsucb 4535

Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4533. (Contributed by NM, 9-Sep-2003.)

Assertion

Ref	Expression
onsucb	|- ( A e. On <-> suc A e. On )

Proof of Theorem onsucb

Step	Hyp	Ref	Expression
1		onsuc 4533	. 2 |- ( A e. On -> suc A e. On )
2		eloni 4406	. . 3 |- ( suc A e. On -> Ord suc A )
3		elex 2771	. . . . 5 |- ( suc A e. On -> suc A e. _V )
4		sucexb 4529	. . . . 5 |- ( A e. _V <-> suc A e. _V )
5	3, 4	sylibr 134	. . . 4 |- ( suc A e. On -> A e. _V )
6		elong 4404	. . . . 5 |- ( A e. _V -> ( A e. On <-> Ord A ) )
7		ordsucg 4534	. . . . 5 |- ( A e. _V -> ( Ord A <-> Ord suc A ) )
8	6, 7	bitrd 188	. . . 4 |- ( A e. _V -> ( A e. On <-> Ord suc A ) )
9	5, 8	syl 14	. . 3 |- ( suc A e. On -> ( A e. On <-> Ord suc A ) )
10	2, 9	mpbird 167	. 2 |- ( suc A e. On -> A e. On )
11	1, 10	impbii 126	1 |- ( A e. On <-> suc A e. On )

Syntax hints:   <-> wb 105   e. wcel 2164   _Vcvv 2760   Ord word 4393   Oncon0 4394   suc csuc 4396

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402

This theorem is referenced by:  onsucmin  4539  onsucuni2  4596