Theorem unizlim 3119

Description: An ordinal equal to its own union is either zero or a limit ordinal.

Assertion

Ref	Expression
unizlim	⊢ (Ord A → (A = ∪A ↔ (A = ∅ ⋁ Lim A)))

Proof of Theorem unizlim

Step	Hyp	Ref	Expression
1		df-lim 2959	. . . . . . . . 9 ⊢ (Lim A ↔ (Ord A ⋀ A ≠ ∅ ⋀ A = ∪A))
2	1	biimpr 152	. . . . . . . 8 ⊢ ((Ord A ⋀ A ≠ ∅ ⋀ A = ∪A) → Lim A)
3	2	3exp 834	. . . . . . 7 ⊢ (Ord A → (A ≠ ∅ → (A = ∪A → Lim A)))
4		df-ne 1590	. . . . . . 7 ⊢ (A ≠ ∅ ↔ ¬ A = ∅)
5	3, 4	syl5ibr 207	. . . . . 6 ⊢ (Ord A → (¬ A = ∅ → (A = ∪A → Lim A)))
6	5	com23 32	. . . . 5 ⊢ (Ord A → (A = ∪A → (¬ A = ∅ → Lim A)))
7	6	imp 350	. . . 4 ⊢ ((Ord A ⋀ A = ∪A) → (¬ A = ∅ → Lim A))
8	7	orrd 233	. . 3 ⊢ ((Ord A ⋀ A = ∪A) → (A = ∅ ⋁ Lim A))
9	8	ex 373	. 2 ⊢ (Ord A → (A = ∪A → (A = ∅ ⋁ Lim A)))
10		uni0 2529	. . . . 5 ⊢ ∪∅ = ∅
11	10	eqcomi 1482	. . . 4 ⊢ ∅ = ∪∅
12		id 59	. . . 4 ⊢ (A = ∅ → A = ∅)
13		unieq 2514	. . . 4 ⊢ (A = ∅ → ∪A = ∪∅)
14	11, 12, 13	3eqtr4a 1535	. . 3 ⊢ (A = ∅ → A = ∪A)
15		limuni 3035	. . 3 ⊢ (Lim A → A = ∪A)
16	14, 15	jaoi 341	. 2 ⊢ ((A = ∅ ⋁ Lim A) → A = ∪A)
17	9, 16	impbid1 519	1 ⊢ (Ord A → (A = ∪A ↔ (A = ∅ ⋁ Lim A)))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋁ wo 222   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ≠ wne 1588  ∅c0 2283  ∪cuni 2507  Ord word 2953  Lim wlim 2955

This theorem is referenced by:  ordzsl 3122

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-uni 2508  df-lim 2959

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