Theorem unizlim 2364

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 2204	. . . . . . . 8 ⊢ (Lim A ↔ (Ord A ∧ ¬ A = ∅ ∧ A = ∪A))
2	1	biimpr 134	. . . . . . 7 ⊢ ((Ord A ∧ ¬ A = ∅ ∧ A = ∪A) → Lim A)
3	2	3exp 611	. . . . . 6 ⊢ (Ord A → (¬ A = ∅ → (A = ∪A → Lim A)))
4	3	com23 32	. . . . 5 ⊢ (Ord A → (A = ∪A → (¬ A = ∅ → Lim A)))
5	4	imp 277	. . . 4 ⊢ ((Ord A ∧ A = ∪A) → (¬ A = ∅ → Lim A))
6	5	orrd 203	. . 3 ⊢ ((Ord A ∧ A = ∪A) → (A = ∅ ∨ Lim A))
7	6	exp 291	. 2 ⊢ (Ord A → (A = ∪A → (A = ∅ ∨ Lim A)))
8		uni0 1938	. . . . . 6 ⊢ ∪∅ = ∅
9	8	cleqcomi 1105	. . . . 5 ⊢ ∅ = ∪∅
10		id 9	. . . . . 6 ⊢ (A = ∅ → A = ∅)
11		unieq 1927	. . . . . 6 ⊢ (A = ∅ → ∪A = ∪∅)
12	10, 11	cleq12d 1115	. . . . 5 ⊢ (A = ∅ → (A = ∪A ↔ ∅ = ∪∅))
13	9, 12	mpbiri 169	. . . 4 ⊢ (A = ∅ → A = ∪A)
14		limuni 2284	. . . 4 ⊢ (Lim A → A = ∪A)
15	13, 14	jaoi 275	. . 3 ⊢ ((A = ∅ ∨ Lim A) → A = ∪A)
16	15	a1i 7	. 2 ⊢ (Ord A → ((A = ∅ ∨ Lim A) → A = ∪A))
17	7, 16	impbid 397	1 ⊢ (Ord A → (A = ∪A ↔ (A = ∅ ∨ Lim A)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∧ w3a 581   = wceq 1091  ∅c0 1707  ∪cuni 1919  Ord word 2198  Lim wlim 2200

This theorem is referenced by:  ordzsl 2366

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-uni 1920  df-lim 2204

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