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Theorem limeq 4140
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 4135 . . 3  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
2 eleq2 2143 . . 3  |-  ( A  =  B  ->  ( (/) 
e.  A  <->  (/)  e.  B
) )
3 id 19 . . . 4  |-  ( A  =  B  ->  A  =  B )
4 unieq 3618 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
53, 4eqeq12d 2096 . . 3  |-  ( A  =  B  ->  ( A  =  U. A  <->  B  =  U. B ) )
61, 2, 53anbi123d 1244 . 2  |-  ( A  =  B  ->  (
( Ord  A  /\  (/) 
e.  A  /\  A  =  U. A )  <->  ( Ord  B  /\  (/)  e.  B  /\  B  =  U. B ) ) )
7 dflim2 4133 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
8 dflim2 4133 . 2  |-  ( Lim 
B  <->  ( Ord  B  /\  (/)  e.  B  /\  B  =  U. B ) )
96, 7, 83bitr4g 221 1  |-  ( A  =  B  ->  ( Lim  A  <->  Lim  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   (/)c0 3258   U.cuni 3609   Ord word 4125   Lim wlim 4127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-in 2980  df-ss 2987  df-uni 3610  df-tr 3884  df-iord 4129  df-ilim 4132
This theorem is referenced by:  limuni2  4160
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