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Theorem elwlim 27896
Description: Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
Assertion
Ref Expression
elwlim  |-  ( X  e. WLim ( R ,  A )  <->  ( X  e.  A  /\  X  =/= 
sup ( A ,  A ,  `' R
)  /\  X  =  sup ( Pred ( R ,  A ,  X
) ,  A ,  R ) ) )

Proof of Theorem elwlim
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 neeq1 2729 . . . 4  |-  ( x  =  X  ->  (
x  =/=  sup ( A ,  A ,  `' R )  <->  X  =/=  sup ( A ,  A ,  `' R ) ) )
2 id 22 . . . . 5  |-  ( x  =  X  ->  x  =  X )
3 predeq3 27765 . . . . . 6  |-  ( x  =  X  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A ,  X ) )
43supeq1d 7799 . . . . 5  |-  ( x  =  X  ->  sup ( Pred ( R ,  A ,  x ) ,  A ,  R )  =  sup ( Pred ( R ,  A ,  X ) ,  A ,  R ) )
52, 4eqeq12d 2473 . . . 4  |-  ( x  =  X  ->  (
x  =  sup ( Pred ( R ,  A ,  x ) ,  A ,  R )  <->  X  =  sup ( Pred ( R ,  A ,  X
) ,  A ,  R ) ) )
61, 5anbi12d 710 . . 3  |-  ( x  =  X  ->  (
( x  =/=  sup ( A ,  A ,  `' R )  /\  x  =  sup ( Pred ( R ,  A ,  x ) ,  A ,  R ) )  <->  ( X  =/=  sup ( A ,  A ,  `' R
)  /\  X  =  sup ( Pred ( R ,  A ,  X
) ,  A ,  R ) ) ) )
7 df-wlim 27886 . . 3  |- WLim ( R ,  A )  =  { x  e.  A  |  ( x  =/= 
sup ( A ,  A ,  `' R
)  /\  x  =  sup ( Pred ( R ,  A ,  x
) ,  A ,  R ) ) }
86, 7elrab2 3218 . 2  |-  ( X  e. WLim ( R ,  A )  <->  ( X  e.  A  /\  ( X  =/=  sup ( A ,  A ,  `' R )  /\  X  =  sup ( Pred ( R ,  A ,  X ) ,  A ,  R ) ) ) )
9 3anass 969 . 2  |-  ( ( X  e.  A  /\  X  =/=  sup ( A ,  A ,  `' R )  /\  X  =  sup ( Pred ( R ,  A ,  X ) ,  A ,  R ) )  <->  ( X  e.  A  /\  ( X  =/=  sup ( A ,  A ,  `' R )  /\  X  =  sup ( Pred ( R ,  A ,  X ) ,  A ,  R ) ) ) )
108, 9bitr4i 252 1  |-  ( X  e. WLim ( R ,  A )  <->  ( X  e.  A  /\  X  =/= 
sup ( A ,  A ,  `' R
)  /\  X  =  sup ( Pred ( R ,  A ,  X
) ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   `'ccnv 4939   supcsup 7793   Predcpred 27760  WLimcwlim 27884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-xp 4946  df-cnv 4948  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-sup 7794  df-pred 27761  df-wlim 27886
This theorem is referenced by: (None)
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