Theorem onsucb 4595

Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4593. (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 4593	. 2 |- ( A e. On -> suc A e. On )
2		eloni 4466	. . 3 |- ( suc A e. On -> Ord suc A )
3		elex 2811	. . . . 5 |- ( suc A e. On -> suc A e. _V )
4		sucexb 4589	. . . . 5 |- ( A e. _V <-> suc A e. _V )
5	3, 4	sylibr 134	. . . 4 |- ( suc A e. On -> A e. _V )
6		elong 4464	. . . . 5 |- ( A e. _V -> ( A e. On <-> Ord A ) )
7		ordsucg 4594	. . . . 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 2799   Ord word 4453   Oncon0 4454   suc csuc 4456

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462

This theorem is referenced by:  onsucmin  4599  onsucuni2  4656