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Theorem limeq 4420
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 4415 . . 3  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
2 neeq1 2467 . . 3  |-  ( A  =  B  ->  ( A  =/=  (/)  <->  B  =/=  (/) ) )
3 id 19 . . . 4  |-  ( A  =  B  ->  A  =  B )
4 unieq 3852 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
53, 4eqeq12d 2310 . . 3  |-  ( A  =  B  ->  ( A  =  U. A  <->  B  =  U. B ) )
61, 2, 53anbi123d 1252 . 2  |-  ( A  =  B  ->  (
( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  <->  ( Ord  B  /\  B  =/=  (/)  /\  B  =  U. B ) ) )
7 df-lim 4413 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
8 df-lim 4413 . 2  |-  ( Lim 
B  <->  ( Ord  B  /\  B  =/=  (/)  /\  B  =  U. B ) )
96, 7, 83bitr4g 279 1  |-  ( A  =  B  ->  ( Lim  A  <->  Lim  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    =/= wne 2459   (/)c0 3468   U.cuni 3843   Ord word 4407   Lim wlim 4409
This theorem is referenced by:  limuni2  4469  0ellim  4470  limuni3  4659  tfinds2  4670  dfom2  4674  limomss  4677  nnlim  4685  limom  4687  ssnlim  4690  onfununi  6374  tfr1a  6426  tz7.44lem1  6434  tz7.44-2  6436  tz7.44-3  6437  oeeulem  6615  limensuc  7054  elom3  7365  r1funlim  7454  rankxplim2  7566  rankxplim3  7567  rankxpsuc  7568  infxpenlem  7657  alephislim  7726  cflim2  7905  winalim  8333  rankcf  8415  gruina  8456  rdgprc0  24221  dfrdg2  24223  dfrdg4  24560  limsucncmpi  24956  limsucncmp  24957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-in 3172  df-ss 3179  df-uni 3844  df-tr 4130  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-lim 4413
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