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Mirrors > Home > MPE Home > Th. List > orduniorsuc | Structured version Visualization version GIF version |
Description: An ordinal class is either its union or the successor of its union. If we adopt the view that zero is a limit ordinal, this means every ordinal class is either a limit or a successor. (Contributed by NM, 13-Sep-2003.) |
Ref | Expression |
---|---|
orduniorsuc | ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orduniss 6460 | . . . . . 6 ⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴) | |
2 | orduni 7779 | . . . . . . . 8 ⊢ (Ord 𝐴 → Ord ∪ 𝐴) | |
3 | ordelssne 6390 | . . . . . . . 8 ⊢ ((Ord ∪ 𝐴 ∧ Ord 𝐴) → (∪ 𝐴 ∈ 𝐴 ↔ (∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴))) | |
4 | 2, 3 | mpancom 684 | . . . . . . 7 ⊢ (Ord 𝐴 → (∪ 𝐴 ∈ 𝐴 ↔ (∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴))) |
5 | 4 | biimprd 247 | . . . . . 6 ⊢ (Ord 𝐴 → ((∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴) → ∪ 𝐴 ∈ 𝐴)) |
6 | 1, 5 | mpand 691 | . . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → ∪ 𝐴 ∈ 𝐴)) |
7 | ordsucss 7808 | . . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 ∈ 𝐴 → suc ∪ 𝐴 ⊆ 𝐴)) | |
8 | 6, 7 | syld 47 | . . . 4 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → suc ∪ 𝐴 ⊆ 𝐴)) |
9 | ordsucuni 7819 | . . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ suc ∪ 𝐴) | |
10 | 8, 9 | jctild 524 | . . 3 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → (𝐴 ⊆ suc ∪ 𝐴 ∧ suc ∪ 𝐴 ⊆ 𝐴))) |
11 | df-ne 2939 | . . . 4 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴) | |
12 | necom 2992 | . . . 4 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ∪ 𝐴 ≠ 𝐴) | |
13 | 11, 12 | bitr3i 276 | . . 3 ⊢ (¬ 𝐴 = ∪ 𝐴 ↔ ∪ 𝐴 ≠ 𝐴) |
14 | eqss 3996 | . . 3 ⊢ (𝐴 = suc ∪ 𝐴 ↔ (𝐴 ⊆ suc ∪ 𝐴 ∧ suc ∪ 𝐴 ⊆ 𝐴)) | |
15 | 10, 13, 14 | 3imtr4g 295 | . 2 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴)) |
16 | 15 | orrd 859 | 1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ⊆ wss 3947 ∪ cuni 4907 Ord word 6362 suc csuc 6365 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6366 df-on 6367 df-suc 6369 |
This theorem is referenced by: onuniorsuc 7827 oeeulem 8603 cantnfp1lem2 9676 cantnflem1 9686 cnfcom2lem 9698 dfac12lem1 10140 dfac12lem2 10141 ttukeylem3 10508 ttukeylem5 10510 ttukeylem6 10511 ordtoplem 35623 ordcmp 35635 onsucuni3 36551 aomclem5 42102 omlimcl2 42293 onov0suclim 42326 dflim5 42381 onsetreclem3 47839 |
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