Metamath Proof Explorer

Theorem elwlim

Description: Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018) (Revised by AV, 10-Oct-2021)

Ref Expression
Assertion elwlim 
|- ( X e. WLim ( R , A ) <-> ( X e. A /\ X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) )

Proof

Step Hyp Ref Expression
1 neeq1 
 |-  ( x = X -> ( x =/= inf ( A , A , R ) <-> X =/= inf ( A , A , R ) ) )
2 id 
 |-  ( x = X -> x = X )
3 predeq3 
 |-  ( x = X -> Pred ( R , A , x ) = Pred ( R , A , X ) )
4 3 supeq1d 
 |-  ( x = X -> sup ( Pred ( R , A , x ) , A , R ) = sup ( Pred ( R , A , X ) , A , R ) )
5 2 4 eqeq12d 
 |-  ( x = X -> ( x = sup ( Pred ( R , A , x ) , A , R ) <-> X = sup ( Pred ( R , A , X ) , A , R ) ) )
6 1 5 anbi12d 
 |-  ( x = X -> ( ( x =/= inf ( A , A , R ) /\ x = sup ( Pred ( R , A , x ) , A , R ) ) <-> ( X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) ) )
7 df-wlim 
 |-  WLim ( R , A ) = { x e. A | ( x =/= inf ( A , A , R ) /\ x = sup ( Pred ( R , A , x ) , A , R ) ) }
8 6 7 elrab2 
 |-  ( X e. WLim ( R , A ) <-> ( X e. A /\ ( X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) ) )
9 3anass 
 |-  ( ( X e. A /\ X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) <-> ( X e. A /\ ( X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) ) )
10 8 9 bitr4i 
 |-  ( X e. WLim ( R , A ) <-> ( X e. A /\ X =/= inf ( A , A , R ) /\ X = sup ( Pred ( R , A , X ) , A , R ) ) )