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

Theorem unizlim 6300
Description: An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
Assertion
Ref Expression
unizlim (Ord 𝐴 → (𝐴 = 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))

Proof of Theorem unizlim
StepHypRef Expression
1 df-ne 3014 . . . . . . 7 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 df-lim 6189 . . . . . . . . 9 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
32biimpri 229 . . . . . . . 8 ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) → Lim 𝐴)
433exp 1111 . . . . . . 7 (Ord 𝐴 → (𝐴 ≠ ∅ → (𝐴 = 𝐴 → Lim 𝐴)))
51, 4syl5bir 244 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = ∅ → (𝐴 = 𝐴 → Lim 𝐴)))
65com23 86 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 → (¬ 𝐴 = ∅ → Lim 𝐴)))
76imp 407 . . . 4 ((Ord 𝐴𝐴 = 𝐴) → (¬ 𝐴 = ∅ → Lim 𝐴))
87orrd 857 . . 3 ((Ord 𝐴𝐴 = 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴))
98ex 413 . 2 (Ord 𝐴 → (𝐴 = 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴)))
10 uni0 4857 . . . . 5 ∅ = ∅
1110eqcomi 2827 . . . 4 ∅ =
12 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
13 unieq 4838 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
1411, 12, 133eqtr4a 2879 . . 3 (𝐴 = ∅ → 𝐴 = 𝐴)
15 limuni 6244 . . 3 (Lim 𝐴𝐴 = 𝐴)
1614, 15jaoi 851 . 2 ((𝐴 = ∅ ∨ Lim 𝐴) → 𝐴 = 𝐴)
179, 16impbid1 226 1 (Ord 𝐴 → (𝐴 = 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wne 3013  c0 4288   cuni 4830  Ord word 6183  Lim wlim 6185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-uni 4831  df-lim 6189
This theorem is referenced by:  ordzsl  7549  oeeulem  8216  cantnfp1lem2  9130  cantnflem1  9140  cnfcom2lem  9152  ordcmp  33692
  Copyright terms: Public domain W3C validator