Theorem unizlim 6508

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 2938	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2		df-lim 6390	. . . . . . . . 9 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
3	2	biimpri 228	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴)
4	3	3exp 1118	. . . . . . 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 863	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴))
9	8	ex 412	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴)))
10		uni0 4939	. . . . 5 ⊢ ∪ ∅ = ∅
11	10	eqcomi 2743	. . . 4 ⊢ ∅ = ∪ ∅
12		id 22	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
13		unieq 4922	. . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
14	11, 12, 13	3eqtr4a 2800	. . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∪ 𝐴)
15		limuni 6446	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
16	14, 15	jaoi 857	. 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 847   ∧ w3a 1086   = wceq 1536   ≠ wne 2937  ∅c0 4338  ∪ cuni 4911  Ord word 6384  Lim wlim 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-v 3479  df-dif 3965  df-ss 3979  df-nul 4339  df-sn 4631  df-uni 4912  df-lim 6390

This theorem is referenced by:  ordzsl  7865  oeeulem  8637  cantnfp1lem2  9716  cantnflem1  9726  cnfcom2lem  9738  ordcmp  36429  onsucf1olem  43259  onov0suclim  43263