Theorem unizlim 6518

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 2947	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		df-lim 6400	. . . . . . . . 9 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
3	2	biimpri 228	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴)
4	3	3exp 1119	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴)))
5	1, 4	biimtrrid 243	. . . . . 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 862	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴))
9	8	ex 412	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴)))
10		uni0 4959	. . . . 5 ⊢ ∪ ∅ = ∅
11	10	eqcomi 2749	. . . 4 ⊢ ∅ = ∪ ∅
12		id 22	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
13		unieq 4942	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
14	11, 12, 13	3eqtr4a 2806	. . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∪ 𝐴)
15		limuni 6456	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
16	14, 15	jaoi 856	. 2 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) → 𝐴 = ∪ 𝐴)
17	9, 16	impbid1 225	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ≠ wne 2946  ∅c0 4352  ∪ cuni 4931  Ord word 6394  Lim wlim 6396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-v 3490  df-dif 3979  df-ss 3993  df-nul 4353  df-sn 4649  df-uni 4932  df-lim 6400

This theorem is referenced by:  ordzsl  7882  oeeulem  8657  cantnfp1lem2  9748  cantnflem1  9758  cnfcom2lem  9770  ordcmp  36413  onsucf1olem  43232  onov0suclim  43236