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Theorem limeq 4337
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 4332 . . 3  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
2 eleq2 2221 . . 3  |-  ( A  =  B  ->  ( (/) 
e.  A  <->  (/)  e.  B
) )
3 id 19 . . . 4  |-  ( A  =  B  ->  A  =  B )
4 unieq 3781 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
53, 4eqeq12d 2172 . . 3  |-  ( A  =  B  ->  ( A  =  U. A  <->  B  =  U. B ) )
61, 2, 53anbi123d 1294 . 2  |-  ( A  =  B  ->  (
( Ord  A  /\  (/) 
e.  A  /\  A  =  U. A )  <->  ( Ord  B  /\  (/)  e.  B  /\  B  =  U. B ) ) )
7 dflim2 4330 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
8 dflim2 4330 . 2  |-  ( Lim 
B  <->  ( Ord  B  /\  (/)  e.  B  /\  B  =  U. B ) )
96, 7, 83bitr4g 222 1  |-  ( A  =  B  ->  ( Lim  A  <->  Lim  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   (/)c0 3394   U.cuni 3772   Ord word 4322   Lim wlim 4324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-in 3108  df-ss 3115  df-uni 3773  df-tr 4063  df-iord 4326  df-ilim 4329
This theorem is referenced by:  limuni2  4357
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