Theorem ordsucg 4538

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 4536	. 2 |- ( Ord A -> Ord suc A )
2		sucidg 4451	. . 3 |- ( A e. _V -> A e. suc A )
3		ordelord 4416	. . . 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 2167   _Vcvv 2763   Ord word 4397   suc csuc 4400

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-uni 3840  df-tr 4132  df-iord 4401  df-suc 4406

This theorem is referenced by:  onsucb  4539