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Mirrors > Home > MPE Home > Th. List > ordzsl | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ordzsl | ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orduninsuc 7832 | . . . . . 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 6488 | . . . . 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 862 | . . 3 ⊢ (Ord 𝐴 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴))) |
6 | 3orass 1091 | . . . 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 233 | . 2 ⊢ (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) |
10 | ord0 6418 | . . . 4 ⊢ Ord ∅ | |
11 | ordeq 6372 | . . . 4 ⊢ (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅)) | |
12 | 10, 11 | mpbiri 258 | . . 3 ⊢ (𝐴 = ∅ → Ord 𝐴) |
13 | onsuc 7799 | . . . . . 6 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
14 | eleq1 2822 | . . . . . 6 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On)) | |
15 | 13, 14 | imbitrrid 245 | . . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On)) |
16 | eloni 6375 | . . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
17 | 15, 16 | syl6com 37 | . . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴)) |
18 | 17 | rexlimiv 3149 | . . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴) |
19 | limord 6425 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
20 | 12, 18, 19 | 3jaoi 1428 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∨ wo 846 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ∅c0 4323 ∪ cuni 4909 Ord word 6364 Oncon0 6365 Lim wlim 6366 suc csuc 6367 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 |
This theorem is referenced by: onzsl 7835 tfrlem16 8393 omeulem1 8582 oaabs2 8648 rankxplim3 9876 rankxpsuc 9877 cardlim 9967 cardaleph 10084 cflim2 10258 dfrdg2 34767 |
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