Theorem unizlim 6478

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

Step	Hyp	Ref	Expression
1		df-ne 2933	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		df-lim 6360	. . . . . . . . 9 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
3	2	biimpri 227	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴)
4	3	3exp 1116	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴)))
5	1, 4	biimtrrid 242	. . . . . 6 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴)))
6	5	com23 86	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (¬ 𝐴 = ∅ → Lim 𝐴)))
7	6	imp 406	. . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (¬ 𝐴 = ∅ → Lim 𝐴))
8	7	orrd 860	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴))
9	8	ex 412	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴)))
10		uni0 4930	. . . . 5 ⊢ ∪ ∅ = ∅
11	10	eqcomi 2733	. . . 4 ⊢ ∅ = ∪ ∅
12		id 22	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
13		unieq 4911	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
14	11, 12, 13	3eqtr4a 2790	. . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∪ 𝐴)
15		limuni 6416	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
16	14, 15	jaoi 854	. 2 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) → 𝐴 = ∪ 𝐴)
17	9, 16	impbid1 224	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ≠ wne 2932  ∅c0 4315  ∪ cuni 4900  Ord word 6354  Lim wlim 6356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958  df-nul 4316  df-sn 4622  df-uni 4901  df-lim 6360

This theorem is referenced by:  ordzsl  7828  oeeulem  8597  cantnfp1lem2  9671  cantnflem1  9681  cnfcom2lem  9693  ordcmp  35823  onsucf1olem  42534  onov0suclim  42538