Theorem ordsucg 4502

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 4500	. 2 |- ( Ord A -> Ord suc A )
2		sucidg 4417	. . 3 |- ( A e. _V -> A e. suc A )
3		ordelord 4382	. . . 4 |- ( ( Ord suc A /\ A e. suc A ) -> Ord A )
4	3	ex 115	. . 3 |- ( Ord suc A -> ( A e. suc A -> Ord A ) )
5	2, 4	syl5com 29	. 2 |- ( A e. _V -> ( Ord suc A -> Ord A ) )
6	1, 5	impbid2 143	1 |- ( A e. _V -> ( Ord A <-> Ord suc A ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2148   _Vcvv 2738   Ord word 4363   suc csuc 4366

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-uni 3811  df-tr 4103  df-iord 4367  df-suc 4372

This theorem is referenced by:  onsucb  4503