Theorem ordzsl 7789

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 7787	. . . . . 6 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2	1	biimprd 250	. . . . 5 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → 𝐴 = ∪ 𝐴))
3		unizlim 6438	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
4	2, 3	sylibd 241	. . . 4 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ Lim 𝐴)))
5	4	orrd 870	. . 3 ⊢ (Ord 𝐴 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
6		3orass 1096	. . . 4 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)))
7		or12 927	. . . 4 ⊢ ((𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
8	6, 7	bitri 277	. . 3 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
9	5, 8	sylibr 236	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
10		ord0 6368	. . . 4 ⊢ Ord ∅
11		ordeq 6321	. . . 4 ⊢ (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅))
12	10, 11	mpbiri 260	. . 3 ⊢ (𝐴 = ∅ → Ord 𝐴)
13		onsuc 7757	. . . . . 6 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
14		eleq1 2829	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
15	13, 14	imbitrrid 248	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On))
16		eloni 6324	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
17	15, 16	syl6com 37	. . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴))
18	17	rexlimiv 3135	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴)
19		limord 6375	. . 3 ⊢ (Lim 𝐴 → Ord 𝐴)
20	12, 18, 19	3jaoi 1437	. 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) → Ord 𝐴)
21	9, 20	impbii 211	1 ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∨ wo 854   ∨ w3o 1092   = wceq 1548   ∈ wcel 2121  ∃wrex 3065  ∅c0 4264  ∪ cuni 4841  Ord word 6313  Oncon0 6314  Lim wlim 6315  suc csuc 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-tr 5183  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320

This theorem is referenced by:  onzsl  7790  tfrlem16  8326  omeulem1  8511  oaabs2  8579  rankxplim3  9800  rankxpsuc  9801  cardlim  9891  cardaleph  10006  cflim2  10180  dfrdg2  36036