Theorem ordelsuc 4541

Description: A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)

Assertion

Ref	Expression
ordelsuc	|- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )

Proof of Theorem ordelsuc

Step	Hyp	Ref	Expression
1		ordsucss 4540	. . 3 |- ( Ord B -> ( A e. B -> suc A C_ B ) )
2	1	adantl 277	. 2 |- ( ( A e. C /\ Ord B ) -> ( A e. B -> suc A C_ B ) )
3		sucssel 4459	. . 3 |- ( A e. C -> ( suc A C_ B -> A e. B ) )
4	3	adantr 276	. 2 |- ( ( A e. C /\ Ord B ) -> ( suc A C_ B -> A e. B ) )
5	2, 4	impbid 129	1 |- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2167   C_ wss 3157   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-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  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:  onsucssi  4542  onsucmin  4543  onsucelsucr  4544  onsucsssucr  4545  onsucsssucexmid  4563  frecsuclem  6464  ordgt0ge1  6493  nnsucsssuc  6550  ennnfonelemk  12617  nninfsellemeq  15658