MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  limeq Structured version   Visualization version   GIF version

Theorem limeq 5879
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 5874 . . 3 (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
2 neeq1 3005 . . 3 (𝐴 = 𝐵 → (𝐴 ≠ ∅ ↔ 𝐵 ≠ ∅))
3 id 22 . . . 4 (𝐴 = 𝐵𝐴 = 𝐵)
4 unieq 4583 . . . 4 (𝐴 = 𝐵 𝐴 = 𝐵)
53, 4eqeq12d 2786 . . 3 (𝐴 = 𝐵 → (𝐴 = 𝐴𝐵 = 𝐵))
61, 2, 53anbi123d 1547 . 2 (𝐴 = 𝐵 → ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) ↔ (Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)))
7 df-lim 5872 . 2 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
8 df-lim 5872 . 2 (Lim 𝐵 ↔ (Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵))
96, 7, 83bitr4g 303 1 (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1071   = wceq 1631  wne 2943  c0 4064   cuni 4575  Ord word 5866  Lim wlim 5868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-in 3731  df-ss 3738  df-uni 4576  df-tr 4888  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-ord 5870  df-lim 5872
This theorem is referenced by:  limuni2  5930  0ellim  5931  limuni3  7200  tfinds2  7211  dfom2  7215  limomss  7218  nnlim  7226  limom  7228  ssnlim  7231  onfununi  7592  tfr1a  7644  tz7.44lem1  7655  tz7.44-2  7657  tz7.44-3  7658  oeeulem  7836  limensuc  8294  elom3  8710  r1funlim  8794  rankxplim2  8908  rankxplim3  8909  rankxpsuc  8910  infxpenlem  9037  alephislim  9107  cflim2  9288  winalim  9720  rankcf  9802  gruina  9843  rdgprc0  32036  dfrdg2  32038  dfrdg4  32396  limsucncmpi  32782  limsucncmp  32783
  Copyright terms: Public domain W3C validator