Theorem onsucb 4599

Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4597. (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 4597	. 2 |- ( A e. On -> suc A e. On )
2		eloni 4470	. . 3 |- ( suc A e. On -> Ord suc A )
3		elex 2812	. . . . 5 |- ( suc A e. On -> suc A e. _V )
4		sucexb 4593	. . . . 5 |- ( A e. _V <-> suc A e. _V )
5	3, 4	sylibr 134	. . . 4 |- ( suc A e. On -> A e. _V )
6		elong 4468	. . . . 5 |- ( A e. _V -> ( A e. On <-> Ord A ) )
7		ordsucg 4598	. . . . 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 2200   _Vcvv 2800   Ord word 4457   Oncon0 4458   suc csuc 4460

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3892  df-tr 4186  df-iord 4461  df-on 4463  df-suc 4466

This theorem is referenced by:  onsucmin  4603  onsucuni2  4660