Theorem ordzsl 2366

Description: An ordinal is zero, a successor ordinal, or a limit ordinal.

Assertion

Ref	Expression
ordzsl	⊢ (Ord A ↔ (A = ∅ ∨ ∃x ∈ On A = suc x ∨ Lim A))

Distinct variable group(s):   x,A

Proof of Theorem ordzsl

Step	Hyp	Ref	Expression
1		orduninsuc 2365	. . . . . 6 ⊢ (Ord A → (A = ∪A ↔ ¬ ∃x ∈ On A = suc x))
2	1	biimprd 136	. . . . 5 ⊢ (Ord A → (¬ ∃x ∈ On A = suc x → A = ∪A))
3		unizlim 2364	. . . . 5 ⊢ (Ord A → (A = ∪A ↔ (A = ∅ ∨ Lim A)))
4	2, 3	sylibd 177	. . . 4 ⊢ (Ord A → (¬ ∃x ∈ On A = suc x → (A = ∅ ∨ Lim A)))
5	4	orrd 203	. . 3 ⊢ (Ord A → (∃x ∈ On A = suc x ∨ (A = ∅ ∨ Lim A)))
6		3orass 584	. . . 4 ⊢ ((A = ∅ ∨ ∃x ∈ On A = suc x ∨ Lim A) ↔ (A = ∅ ∨ (∃x ∈ On A = suc x ∨ Lim A)))
7		or12 217	. . . 4 ⊢ ((A = ∅ ∨ (∃x ∈ On A = suc x ∨ Lim A)) ↔ (∃x ∈ On A = suc x ∨ (A = ∅ ∨ Lim A)))
8	6, 7	bitr 151	. . 3 ⊢ ((A = ∅ ∨ ∃x ∈ On A = suc x ∨ Lim A) ↔ (∃x ∈ On A = suc x ∨ (A = ∅ ∨ Lim A)))
9	5, 8	sylibr 175	. 2 ⊢ (Ord A → (A = ∅ ∨ ∃x ∈ On A = suc x ∨ Lim A))
10		ord0 2276	. . . 4 ⊢ Ord ∅
11		ordeq 2206	. . . 4 ⊢ (A = ∅ → (Ord A ↔ Ord ∅))
12	10, 11	mpbiri 169	. . 3 ⊢ (A = ∅ → Ord A)
13		eleq1 1149	. . . . . . 7 ⊢ (A = suc x → (A ∈ On ↔ suc x ∈ On))
14		suceloni 2314	. . . . . . 7 ⊢ (x ∈ On → suc x ∈ On)
15	13, 14	syl5bir 184	. . . . . 6 ⊢ (A = suc x → (x ∈ On → A ∈ On))
16	15	com12 13	. . . . 5 ⊢ (x ∈ On → (A = suc x → A ∈ On))
17		eloni 2209	. . . . 5 ⊢ (A ∈ On → Ord A)
18	16, 17	syl6 23	. . . 4 ⊢ (x ∈ On → (A = suc x → Ord A))
19	18	r19.23aiv 1284	. . 3 ⊢ (∃x ∈ On A = suc x → Ord A)
20		limord 2283	. . 3 ⊢ (Lim A → Ord A)
21	12, 19, 20	3jaoi 633	. 2 ⊢ ((A = ∅ ∨ ∃x ∈ On A = suc x ∨ Lim A) → Ord A)
22	9, 21	impbi 139	1 ⊢ (Ord A ↔ (A = ∅ ∨ ∃x ∈ On A = suc x ∨ Lim A))

Syntax hints:  ¬ wn 1   ↔ wb 127   ∨ wo 195   ∨ w3o 580   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ∅c0 1707  ∪cuni 1919  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201

This theorem is referenced by:  onzsl 2367  dflim3 2368  rankr1 3518  cardlim 3657  cardaleph 3690

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205

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