Theorem ordzsl 7692

Description: An ordinal is zero, a successor ordinal, or a limit ordinal. (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 7690	. . . . . 6 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2	1	biimprd 247	. . . . 5 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → 𝐴 = ∪ 𝐴))
3		unizlim 6383	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
4	2, 3	sylibd 238	. . . 4 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ Lim 𝐴)))
5	4	orrd 860	. . 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 918	. . . 4 ⊢ ((𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
8	6, 7	bitri 274	. . 3 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
9	5, 8	sylibr 233	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
10		ord0 6318	. . . 4 ⊢ Ord ∅
11		ordeq 6273	. . . 4 ⊢ (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅))
12	10, 11	mpbiri 257	. . 3 ⊢ (𝐴 = ∅ → Ord 𝐴)
13		suceloni 7659	. . . . . 6 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
14		eleq1 2826	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
15	13, 14	syl5ibr 245	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On))
16		eloni 6276	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
17	15, 16	syl6com 37	. . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴))
18	17	rexlimiv 3209	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴)
19		limord 6325	. . 3 ⊢ (Lim 𝐴 → Ord 𝐴)
20	12, 18, 19	3jaoi 1426	. 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) → Ord 𝐴)
21	9, 20	impbii 208	1 ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∨ wo 844   ∨ w3o 1085   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  ∅c0 4256  ∪ cuni 4839  Ord word 6265  Oncon0 6266  Lim wlim 6267  suc csuc 6268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272

This theorem is referenced by:  onzsl  7693  tfrlem16  8224  omeulem1  8413  oaabs2  8479  rankxplim3  9639  rankxpsuc  9640  cardlim  9730  cardaleph  9845  cflim2  10019  dfrdg2  33771