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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 7043 | . . . . . 6 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) | |
2 | 1 | biimprd 238 | . . . . 5 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → 𝐴 = ∪ 𝐴)) |
3 | unizlim 5844 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴))) | |
4 | 2, 3 | sylibd 229 | . . . 4 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ Lim 𝐴))) |
5 | 4 | orrd 393 | . . 3 ⊢ (Ord 𝐴 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴))) |
6 | 3orass 1040 | . . . 4 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))) | |
7 | or12 545 | . . . 4 ⊢ ((𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴))) | |
8 | 6, 7 | bitri 264 | . . 3 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴))) |
9 | 5, 8 | sylibr 224 | . 2 ⊢ (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) |
10 | ord0 5777 | . . . 4 ⊢ Ord ∅ | |
11 | ordeq 5730 | . . . 4 ⊢ (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅)) | |
12 | 10, 11 | mpbiri 248 | . . 3 ⊢ (𝐴 = ∅ → Ord 𝐴) |
13 | suceloni 7013 | . . . . . 6 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
14 | eleq1 2689 | . . . . . 6 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On)) | |
15 | 13, 14 | syl5ibr 236 | . . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On)) |
16 | eloni 5733 | . . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
17 | 15, 16 | syl6com 37 | . . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴)) |
18 | 17 | rexlimiv 3027 | . . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴) |
19 | limord 5784 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
20 | 12, 18, 19 | 3jaoi 1391 | . 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) → Ord 𝐴) |
21 | 9, 20 | impbii 199 | 1 ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∨ wo 383 ∨ w3o 1036 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∅c0 3915 ∪ cuni 4436 Ord word 5722 Oncon0 5723 Lim wlim 5724 suc csuc 5725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 |
This theorem is referenced by: onzsl 7046 tfrlem16 7489 omeulem1 7662 oaabs2 7725 rankxplim3 8744 rankxpsuc 8745 cardlim 8798 cardaleph 8912 cflim2 9085 dfrdg2 31701 |
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