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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. (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 7600 | . . . . . 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 6308 | . . . . 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 863 | . . 3 ⊢ (Ord 𝐴 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴))) |
6 | 3orass 1092 | . . . 4 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))) | |
7 | or12 921 | . . . 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 6243 | . . . 4 ⊢ Ord ∅ | |
11 | ordeq 6198 | . . . 4 ⊢ (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅)) | |
12 | 10, 11 | mpbiri 261 | . . 3 ⊢ (𝐴 = ∅ → Ord 𝐴) |
13 | suceloni 7570 | . . . . . 6 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
14 | eleq1 2818 | . . . . . 6 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On)) | |
15 | 13, 14 | syl5ibr 249 | . . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On)) |
16 | eloni 6201 | . . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
17 | 15, 16 | syl6com 37 | . . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴)) |
18 | 17 | rexlimiv 3189 | . . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴) |
19 | limord 6250 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
20 | 12, 18, 19 | 3jaoi 1429 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∨ wo 847 ∨ w3o 1088 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 ∅c0 4223 ∪ cuni 4805 Ord word 6190 Oncon0 6191 Lim wlim 6192 suc csuc 6193 |
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 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 |
This theorem is referenced by: onzsl 7603 tfrlem16 8107 omeulem1 8288 oaabs2 8352 rankxplim3 9462 rankxpsuc 9463 cardlim 9553 cardaleph 9668 cflim2 9842 dfrdg2 33441 |
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