Theorem onsucb 4569

Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4567. (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 4567	. 2 |- ( A e. On -> suc A e. On )
2		eloni 4440	. . 3 |- ( suc A e. On -> Ord suc A )
3		elex 2788	. . . . 5 |- ( suc A e. On -> suc A e. _V )
4		sucexb 4563	. . . . 5 |- ( A e. _V <-> suc A e. _V )
5	3, 4	sylibr 134	. . . 4 |- ( suc A e. On -> A e. _V )
6		elong 4438	. . . . 5 |- ( A e. _V -> ( A e. On <-> Ord A ) )
7		ordsucg 4568	. . . . 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 2178   _Vcvv 2776   Ord word 4427   Oncon0 4428   suc csuc 4430

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433  df-suc 4436

This theorem is referenced by:  onsucmin  4573  onsucuni2  4630