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Theorem limeq 4585
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 4580 . . 3  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
2 neeq1 2606 . . 3  |-  ( A  =  B  ->  ( A  =/=  (/)  <->  B  =/=  (/) ) )
3 id 20 . . . 4  |-  ( A  =  B  ->  A  =  B )
4 unieq 4016 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
53, 4eqeq12d 2449 . . 3  |-  ( A  =  B  ->  ( A  =  U. A  <->  B  =  U. B ) )
61, 2, 53anbi123d 1254 . 2  |-  ( A  =  B  ->  (
( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  <->  ( Ord  B  /\  B  =/=  (/)  /\  B  =  U. B ) ) )
7 df-lim 4578 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
8 df-lim 4578 . 2  |-  ( Lim 
B  <->  ( Ord  B  /\  B  =/=  (/)  /\  B  =  U. B ) )
96, 7, 83bitr4g 280 1  |-  ( A  =  B  ->  ( Lim  A  <->  Lim  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    =/= wne 2598   (/)c0 3620   U.cuni 4007   Ord word 4572   Lim wlim 4574
This theorem is referenced by:  limuni2  4634  0ellim  4635  limuni3  4824  tfinds2  4835  dfom2  4839  limomss  4842  nnlim  4850  limom  4852  ssnlim  4855  onfununi  6595  tfr1a  6647  tz7.44lem1  6655  tz7.44-2  6657  tz7.44-3  6658  oeeulem  6836  limensuc  7276  elom3  7595  r1funlim  7684  rankxplim2  7796  rankxplim3  7797  rankxpsuc  7798  infxpenlem  7887  alephislim  7956  cflim2  8135  winalim  8562  rankcf  8644  gruina  8685  rdgprc0  25413  dfrdg2  25415  dfrdg4  25787  limsucncmpi  26187  limsucncmp  26188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-in 3319  df-ss 3326  df-uni 4008  df-tr 4295  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-lim 4578
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