Theorem ordsucg 4568

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 4566	. 2 |- ( Ord A -> Ord suc A )
2		sucidg 4481	. . 3 |- ( A e. _V -> A e. suc A )
3		ordelord 4446	. . . 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 2178   _Vcvv 2776   Ord word 4427   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-ext 2189

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-sn 3649  df-uni 3865  df-tr 4159  df-iord 4431  df-suc 4436

This theorem is referenced by:  onsucb  4569