Theorem onsucb 4514

Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4512. (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 4512	. 2 |- ( A e. On -> suc A e. On )
2		eloni 4387	. . 3 |- ( suc A e. On -> Ord suc A )
3		elex 2760	. . . . 5 |- ( suc A e. On -> suc A e. _V )
4		sucexb 4508	. . . . 5 |- ( A e. _V <-> suc A e. _V )
5	3, 4	sylibr 134	. . . 4 |- ( suc A e. On -> A e. _V )
6		elong 4385	. . . . 5 |- ( A e. _V -> ( A e. On <-> Ord A ) )
7		ordsucg 4513	. . . . 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 2158   _Vcvv 2749   Ord word 4374   Oncon0 4375   suc csuc 4377

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383

This theorem is referenced by:  onsucmin  4518  onsucuni2  4575