Theorem ordzsl 7824

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 7822	. . . . . 6 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2	1	biimprd 248	. . . . 5 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → 𝐴 = ∪ 𝐴))
3		unizlim 6460	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
4	2, 3	sylibd 239	. . . 4 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ Lim 𝐴)))
5	4	orrd 863	. . 3 ⊢ (Ord 𝐴 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
6		3orass 1089	. . . 4 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)))
7		or12 920	. . . 4 ⊢ ((𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
8	6, 7	bitri 275	. . 3 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
9	5, 8	sylibr 234	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
10		ord0 6389	. . . 4 ⊢ Ord ∅
11		ordeq 6342	. . . 4 ⊢ (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅))
12	10, 11	mpbiri 258	. . 3 ⊢ (𝐴 = ∅ → Ord 𝐴)
13		onsuc 7790	. . . . . 6 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
14		eleq1 2817	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
15	13, 14	imbitrrid 246	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On))
16		eloni 6345	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
17	15, 16	syl6com 37	. . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴))
18	17	rexlimiv 3128	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴)
19		limord 6396	. . 3 ⊢ (Lim 𝐴 → Ord 𝐴)
20	12, 18, 19	3jaoi 1430	. 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) → Ord 𝐴)
21	9, 20	impbii 209	1 ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∨ wo 847   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  ∃wrex 3054  ∅c0 4299  ∪ cuni 4874  Ord word 6334  Oncon0 6335  Lim wlim 6336  suc csuc 6337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341

This theorem is referenced by:  onzsl  7825  tfrlem16  8364  omeulem1  8549  oaabs2  8616  rankxplim3  9841  rankxpsuc  9842  cardlim  9932  cardaleph  10049  cflim2  10223  dfrdg2  35790