Theorem ordzsl 7844

Description: An ordinal is zero, a successor ordinal, or a limit ordinal. Remark 1.12 of [Schloeder] p. 2. (Contributed by NM, 1-Oct-2003.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem ordzsl

Step	Hyp	Ref	Expression
1		orduninsuc 7842	. . . . . 6 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2	1	biimprd 251	. . . . 5 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → 𝐴 = ∪ 𝐴))
3		unizlim 6486	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
4	2, 3	sylibd 242	. . . 4 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ Lim 𝐴)))
5	4	orrd 877	. . 3 ⊢ (Ord 𝐴 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
6		3orass 1106	. . . 4 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)))
7		or12 934	. . . 4 ⊢ ((𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
8	6, 7	bitri 278	. . 3 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
9	5, 8	sylibr 237	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
10		ord0 6416	. . . 4 ⊢ Ord ∅
11		ordeq 6368	. . . 4 ⊢ (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅))
12	10, 11	mpbiri 261	. . 3 ⊢ (𝐴 = ∅ → Ord 𝐴)
13		onsuc 7812	. . . . . 6 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
14		eleq1 2850	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
15	13, 14	imbitrrid 249	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On))
16		eloni 6371	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
17	15, 16	syl6com 38	. . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴))
18	17	rexlimiv 3158	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴)
19		limord 6423	. . 3 ⊢ (Lim 𝐴 → Ord 𝐴)
20	12, 18, 19	3jaoi 1454	. 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) → Ord 𝐴)
21	9, 20	impbii 212	1 ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  ∅c0 4282  ∪ cuni 4870  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367

This theorem is used by:  onzsl  7845  tfrlem16  8385  omeulem1  8572  oaabs2  8640  rankxplim3  9866  rankxpsuc  9867  cardlim  9980  cardaleph  10095  cflim2  10268  dfrdg2  36359