Theorem ordsucg 4629

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 4627	. 2 |- ( Ord A -> Ord suc A )
2		sucidg 4542	. . 3 |- ( A e. _V -> A e. suc A )
3		ordelord 4507	. . . 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 2205   _Vcvv 2815   Ord word 4488   suc csuc 4491

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-uni 3920  df-tr 4214  df-iord 4492  df-suc 4497

This theorem is referenced by:  onsucb  4630