Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elwlim Structured version   Unicode version

Theorem elwlim 29622
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 2735 . . . 4  |-  ( x  =  X  ->  (
x  =/=  sup ( A ,  A ,  `' R )  <->  X  =/=  sup ( A ,  A ,  `' R ) ) )
2 id 22 . . . . 5  |-  ( x  =  X  ->  x  =  X )
3 predeq3 29491 . . . . . 6  |-  ( x  =  X  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A ,  X ) )
43supeq1d 7897 . . . . 5  |-  ( x  =  X  ->  sup ( Pred ( R ,  A ,  x ) ,  A ,  R )  =  sup ( Pred ( R ,  A ,  X ) ,  A ,  R ) )
52, 4eqeq12d 2476 . . . 4  |-  ( x  =  X  ->  (
x  =  sup ( Pred ( R ,  A ,  x ) ,  A ,  R )  <->  X  =  sup ( Pred ( R ,  A ,  X
) ,  A ,  R ) ) )
61, 5anbi12d 708 . . 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 29612 . . 3  |- WLim ( R ,  A )  =  { x  e.  A  |  ( x  =/= 
sup ( A ,  A ,  `' R
)  /\  x  =  sup ( Pred ( R ,  A ,  x
) ,  A ,  R ) ) }
86, 7elrab2 3256 . 2  |-  ( X  e. WLim ( R ,  A )  <->  ( X  e.  A  /\  ( X  =/=  sup ( A ,  A ,  `' R )  /\  X  =  sup ( Pred ( R ,  A ,  X ) ,  A ,  R ) ) ) )
9 3anass 975 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   `'ccnv 4987   supcsup 7892   Predcpred 29486  WLimcwlim 29610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-sup 7893  df-pred 29487  df-wlim 29612
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator