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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 6481 | . . . . . 6 ⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴) | |
| 2 | orduni 7809 | . . . . . . . 8 ⊢ (Ord 𝐴 → Ord ∪ 𝐴) | |
| 3 | ordelssne 6411 | . . . . . . . 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 7838 | . . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 ∈ 𝐴 → suc ∪ 𝐴 ⊆ 𝐴)) | |
| 8 | 6, 7 | syld 47 | . . . 4 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → suc ∪ 𝐴 ⊆ 𝐴)) |
| 9 | ordsucuni 7849 | . . . 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 2941 | . . . 4 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴) | |
| 12 | necom 2994 | . . . 4 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ∪ 𝐴 ≠ 𝐴) | |
| 13 | 11, 12 | bitr3i 277 | . . 3 ⊢ (¬ 𝐴 = ∪ 𝐴 ↔ ∪ 𝐴 ≠ 𝐴) |
| 14 | eqss 3999 | . . 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 864 | 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 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ∪ cuni 4907 Ord word 6383 suc csuc 6386 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 |
| This theorem is referenced by: onuniorsuc 7857 oeeulem 8639 cantnfp1lem2 9719 cantnflem1 9729 cnfcom2lem 9741 dfac12lem1 10184 dfac12lem2 10185 ttukeylem3 10551 ttukeylem5 10553 ttukeylem6 10554 ordtoplem 36436 ordcmp 36448 onsucuni3 37368 aomclem5 43070 omlimcl2 43254 onov0suclim 43287 dflim5 43342 onsetreclem3 49226 |
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