Theorem unizlim 6487

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 2956	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		df-lim 6367	. . . . . . . . 9 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
3	2	biimpri 231	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴)
4	3	3exp 1137	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴)))
5	1, 4	biimtrrid 246	. . . . . 6 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴)))
6	5	com23 87	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (¬ 𝐴 = ∅ → Lim 𝐴)))
7	6	imp 412	. . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (¬ 𝐴 = ∅ → Lim 𝐴))
8	7	orrd 877	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴))
9	8	ex 418	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴)))
10		uni0 4895	. . . . 5 ⊢ ∪ ∅ = ∅
11	10	eqcomi 2769	. . . 4 ⊢ ∅ = ∪ ∅
12		id 23	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
13		unieq 4877	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
14	11, 12, 13	3eqtr4a 2821	. . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∪ 𝐴)
15		limuni 6425	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
16	14, 15	jaoi 871	. 2 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) → 𝐴 = ∪ 𝐴)
17	9, 16	impbid1 228	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ≠ wne 2955  ∅c0 4278  ∪ cuni 4866  Ord word 6361  Lim wlim 6363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-ss 3896  df-dif 4047  df-nul 4279  df-uni 4867  df-lim 6367

This theorem is used by:  ordzsl  7857  oeeulem  8614  cantnfp1lem2  9684  cantnflem1  9694  cnfcom2lem  9706  ordcmp  37328  onsucf1olem  44366  onov0suclim  44370