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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 6482 | . . . . . 6 ⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴) | |
2 | orduni 7808 | . . . . . . . 8 ⊢ (Ord 𝐴 → Ord ∪ 𝐴) | |
3 | ordelssne 6412 | . . . . . . . 8 ⊢ ((Ord ∪ 𝐴 ∧ Ord 𝐴) → (∪ 𝐴 ∈ 𝐴 ↔ (∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴))) | |
4 | 2, 3 | mpancom 688 | . . . . . . 7 ⊢ (Ord 𝐴 → (∪ 𝐴 ∈ 𝐴 ↔ (∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴))) |
5 | 4 | biimprd 248 | . . . . . 6 ⊢ (Ord 𝐴 → ((∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴) → ∪ 𝐴 ∈ 𝐴)) |
6 | 1, 5 | mpand 695 | . . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → ∪ 𝐴 ∈ 𝐴)) |
7 | ordsucss 7837 | . . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 ∈ 𝐴 → suc ∪ 𝐴 ⊆ 𝐴)) | |
8 | 6, 7 | syld 47 | . . . 4 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → suc ∪ 𝐴 ⊆ 𝐴)) |
9 | ordsucuni 7848 | . . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ suc ∪ 𝐴) | |
10 | 8, 9 | jctild 525 | . . 3 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → (𝐴 ⊆ suc ∪ 𝐴 ∧ suc ∪ 𝐴 ⊆ 𝐴))) |
11 | df-ne 2938 | . . . 4 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴) | |
12 | necom 2991 | . . . 4 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ∪ 𝐴 ≠ 𝐴) | |
13 | 11, 12 | bitr3i 277 | . . 3 ⊢ (¬ 𝐴 = ∪ 𝐴 ↔ ∪ 𝐴 ≠ 𝐴) |
14 | eqss 4010 | . . 3 ⊢ (𝐴 = suc ∪ 𝐴 ↔ (𝐴 ⊆ suc ∪ 𝐴 ∧ suc ∪ 𝐴 ⊆ 𝐴)) | |
15 | 10, 13, 14 | 3imtr4g 296 | . 2 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴)) |
16 | 15 | orrd 863 | 1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ⊆ wss 3962 ∪ cuni 4911 Ord word 6384 suc csuc 6387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-on 6389 df-suc 6391 |
This theorem is referenced by: onuniorsuc 7856 oeeulem 8637 cantnfp1lem2 9716 cantnflem1 9726 cnfcom2lem 9738 dfac12lem1 10181 dfac12lem2 10182 ttukeylem3 10548 ttukeylem5 10550 ttukeylem6 10551 ordtoplem 36417 ordcmp 36429 onsucuni3 37349 aomclem5 43046 omlimcl2 43230 onov0suclim 43263 dflim5 43318 onsetreclem3 48937 |
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