Theorem ordsucg 4228

Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)

Assertion

Ref	Expression
ordsucg	|- ( A e. _V -> ( Ord A <-> Ord suc A ) )

Proof of Theorem ordsucg

Step	Hyp	Ref	Expression
1		ordsucim 4226	. 2 |- ( Ord A -> Ord suc A )
2		sucidg 4153	. . 3 |- ( A e. _V -> A e. suc A )
3		ordelord 4118	. . . 4 |- ( ( Ord suc A /\ A e. suc A ) -> Ord A )
4	3	ex 108	. . 3 |- ( Ord suc A -> ( A e. suc A -> Ord A ) )
5	2, 4	syl5com 26	. 2 |- ( A e. _V -> ( Ord suc A -> Ord A ) )
6	1, 5	impbid2 131	1 |- ( A e. _V -> ( Ord A <-> Ord suc A ) )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   _Vcvv 2557   Ord word 4099   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103  df-suc 4108

This theorem is referenced by:  sucelon  4229