Theorem unizlim 6489

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 2961	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		df-lim 6369	. . . . . . . . 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 4903	. . . . 5 ⊢ ∪ ∅ = ∅
11	10	eqcomi 2774	. . . 4 ⊢ ∅ = ∪ ∅
12		id 23	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
13		unieq 4885	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
14	11, 12, 13	3eqtr4a 2826	. . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∪ 𝐴)
15		limuni 6427	. . 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 1570   ≠ wne 2960  ∅c0 4286  ∪ cuni 4874  Ord word 6363  Lim wlim 6365

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3909  df-ss 3923  df-nul 4287  df-uni 4875  df-lim 6369

This theorem is used by:  ordzsl  7843  oeeulem  8589  cantnfp1lem2  9651  cantnflem1  9661  cnfcom2lem  9673  ordcmp  36991  onsucf1olem  44030  onov0suclim  44034