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Theorem unizlim 6024
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 2938 . . . . . . 7 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 df-lim 5913 . . . . . . . . 9 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
32biimpri 219 . . . . . . . 8 ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) → Lim 𝐴)
433exp 1148 . . . . . . 7 (Ord 𝐴 → (𝐴 ≠ ∅ → (𝐴 = 𝐴 → Lim 𝐴)))
51, 4syl5bir 234 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = ∅ → (𝐴 = 𝐴 → Lim 𝐴)))
65com23 86 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 → (¬ 𝐴 = ∅ → Lim 𝐴)))
76imp 395 . . . 4 ((Ord 𝐴𝐴 = 𝐴) → (¬ 𝐴 = ∅ → Lim 𝐴))
87orrd 889 . . 3 ((Ord 𝐴𝐴 = 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴))
98ex 401 . 2 (Ord 𝐴 → (𝐴 = 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴)))
10 uni0 4623 . . . . 5 ∅ = ∅
1110eqcomi 2774 . . . 4 ∅ =
12 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
13 unieq 4602 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
1411, 12, 133eqtr4a 2825 . . 3 (𝐴 = ∅ → 𝐴 = 𝐴)
15 limuni 5968 . . 3 (Lim 𝐴𝐴 = 𝐴)
1614, 15jaoi 883 . 2 ((𝐴 = ∅ ∨ Lim 𝐴) → 𝐴 = 𝐴)
179, 16impbid1 216 1 (Ord 𝐴 → (𝐴 = 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wne 2937  c0 4079   cuni 4594  Ord word 5907  Lim wlim 5909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3735  df-in 3739  df-ss 3746  df-nul 4080  df-sn 4335  df-uni 4595  df-lim 5913
This theorem is referenced by:  ordzsl  7243  oeeulem  7886  cantnfp1lem2  8791  cantnflem1  8801  cnfcom2lem  8813  ordcmp  32885
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