ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  limeq Unicode version

Theorem limeq 4362
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 4357 . . 3  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
2 eleq2 2234 . . 3  |-  ( A  =  B  ->  ( (/) 
e.  A  <->  (/)  e.  B
) )
3 id 19 . . . 4  |-  ( A  =  B  ->  A  =  B )
4 unieq 3805 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
53, 4eqeq12d 2185 . . 3  |-  ( A  =  B  ->  ( A  =  U. A  <->  B  =  U. B ) )
61, 2, 53anbi123d 1307 . 2  |-  ( A  =  B  ->  (
( Ord  A  /\  (/) 
e.  A  /\  A  =  U. A )  <->  ( Ord  B  /\  (/)  e.  B  /\  B  =  U. B ) ) )
7 dflim2 4355 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
8 dflim2 4355 . 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 973    = wceq 1348    e. wcel 2141   (/)c0 3414   U.cuni 3796   Ord word 4347   Lim wlim 4349
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-ilim 4354
This theorem is referenced by:  limuni2  4382
  Copyright terms: Public domain W3C validator