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Theorem limeq 6396
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 6391 . . 3 (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
2 neeq1 3003 . . 3 (𝐴 = 𝐵 → (𝐴 ≠ ∅ ↔ 𝐵 ≠ ∅))
3 id 22 . . . 4 (𝐴 = 𝐵𝐴 = 𝐵)
4 unieq 4918 . . . 4 (𝐴 = 𝐵 𝐴 = 𝐵)
53, 4eqeq12d 2753 . . 3 (𝐴 = 𝐵 → (𝐴 = 𝐴𝐵 = 𝐵))
61, 2, 53anbi123d 1438 . 2 (𝐴 = 𝐵 → ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) ↔ (Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)))
7 df-lim 6389 . 2 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
8 df-lim 6389 . 2 (Lim 𝐵 ↔ (Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵))
96, 7, 83bitr4g 314 1 (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wne 2940  c0 4333   cuni 4907  Ord word 6383  Lim wlim 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-lim 6389
This theorem is referenced by:  limuni2  6446  0ellim  6447  limuni3  7873  tfinds2  7885  dfom2  7889  limomss  7892  nnlim  7901  limom  7903  ssnlim  7907  onfununi  8381  tfr1a  8434  tz7.44lem1  8445  tz7.44-2  8447  tz7.44-3  8448  1ellim  8536  2ellim  8537  oeeulem  8639  limensuc  9194  elom3  9688  r1funlim  9806  rankxplim2  9920  rankxplim3  9921  rankxpsuc  9922  infxpenlem  10053  alephislim  10123  cflim2  10303  winalim  10735  rankcf  10817  gruina  10858  scutbdaybnd2lim  27862  rdgprc0  35794  dfrdg2  35796  dfrdg4  35952  limsucncmpi  36446  limsucncmp  36447  omlimcl2  43254  onexlimgt  43255  onov0suclim  43287  succlg  43341  dflim5  43342  nlim1NEW  43455  nlim2NEW  43456  nlim3  43457  nlim4  43458  dfsucon  43536
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