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Theorem limeq 4899
Description: Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
limeq  |-  ( A  =  B  ->  ( Lim  A  <->  Lim  B ) )

Proof of Theorem limeq
StepHypRef Expression
1 ordeq 4894 . . 3  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
2 neeq1 2738 . . 3  |-  ( A  =  B  ->  ( A  =/=  (/)  <->  B  =/=  (/) ) )
3 id 22 . . . 4  |-  ( A  =  B  ->  A  =  B )
4 unieq 4259 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
53, 4eqeq12d 2479 . . 3  |-  ( A  =  B  ->  ( A  =  U. A  <->  B  =  U. B ) )
61, 2, 53anbi123d 1299 . 2  |-  ( A  =  B  ->  (
( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  <->  ( Ord  B  /\  B  =/=  (/)  /\  B  =  U. B ) ) )
7 df-lim 4892 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
8 df-lim 4892 . 2  |-  ( Lim 
B  <->  ( Ord  B  /\  B  =/=  (/)  /\  B  =  U. B ) )
96, 7, 83bitr4g 288 1  |-  ( A  =  B  ->  ( Lim  A  <->  Lim  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1395    =/= wne 2652   (/)c0 3793   U.cuni 4251   Ord word 4886   Lim wlim 4888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-in 3478  df-ss 3485  df-uni 4252  df-tr 4551  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-lim 4892
This theorem is referenced by:  limuni2  4948  0ellim  4949  limuni3  6686  tfinds2  6697  dfom2  6701  limomss  6704  nnlim  6712  limom  6714  ssnlim  6717  onfununi  7030  tfr1a  7081  tz7.44lem1  7089  tz7.44-2  7091  tz7.44-3  7092  oeeulem  7268  limensuc  7713  elom3  8082  r1funlim  8201  rankxplim2  8315  rankxplim3  8316  rankxpsuc  8317  infxpenlem  8408  alephislim  8481  cflim2  8660  winalim  9090  rankcf  9172  gruina  9213  rdgprc0  29443  dfrdg2  29445  dfrdg4  29805  limsucncmpi  30115  limsucncmp  30116
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