Metamath Proof Explorer


Theorem ordtoplem

Description: Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015)

Ref Expression
Hypothesis ordtoplem.1
|- ( U. A e. On -> suc U. A e. S )
Assertion ordtoplem
|- ( Ord A -> ( A =/= U. A -> A e. S ) )

Proof

Step Hyp Ref Expression
1 ordtoplem.1
 |-  ( U. A e. On -> suc U. A e. S )
2 df-ne
 |-  ( A =/= U. A <-> -. A = U. A )
3 ordeleqon
 |-  ( Ord A <-> ( A e. On \/ A = On ) )
4 unon
 |-  U. On = On
5 4 eqcomi
 |-  On = U. On
6 id
 |-  ( A = On -> A = On )
7 unieq
 |-  ( A = On -> U. A = U. On )
8 5 6 7 3eqtr4a
 |-  ( A = On -> A = U. A )
9 8 orim2i
 |-  ( ( A e. On \/ A = On ) -> ( A e. On \/ A = U. A ) )
10 3 9 sylbi
 |-  ( Ord A -> ( A e. On \/ A = U. A ) )
11 10 orcomd
 |-  ( Ord A -> ( A = U. A \/ A e. On ) )
12 11 ord
 |-  ( Ord A -> ( -. A = U. A -> A e. On ) )
13 orduniorsuc
 |-  ( Ord A -> ( A = U. A \/ A = suc U. A ) )
14 13 ord
 |-  ( Ord A -> ( -. A = U. A -> A = suc U. A ) )
15 onuni
 |-  ( A e. On -> U. A e. On )
16 eleq1a
 |-  ( suc U. A e. S -> ( A = suc U. A -> A e. S ) )
17 15 1 16 3syl
 |-  ( A e. On -> ( A = suc U. A -> A e. S ) )
18 12 14 17 syl6c
 |-  ( Ord A -> ( -. A = U. A -> A e. S ) )
19 2 18 syl5bi
 |-  ( Ord A -> ( A =/= U. A -> A e. S ) )