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| Mirrors > Home > MPE Home > Th. List > unizlim | Structured version Visualization version GIF version | ||
| Description: An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.) |
| Ref | Expression |
|---|---|
| unizlim | ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ne 2941 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
| 2 | df-lim 6389 | . . . . . . . . 9 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
| 3 | 2 | biimpri 228 | . . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴) |
| 4 | 3 | 3exp 1120 | . . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴))) |
| 5 | 1, 4 | biimtrrid 243 | . . . . . 6 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴))) |
| 6 | 5 | com23 86 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (¬ 𝐴 = ∅ → Lim 𝐴))) |
| 7 | 6 | imp 406 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (¬ 𝐴 = ∅ → Lim 𝐴)) |
| 8 | 7 | orrd 864 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴)) |
| 9 | 8 | ex 412 | . 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴))) |
| 10 | uni0 4935 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
| 11 | 10 | eqcomi 2746 | . . . 4 ⊢ ∅ = ∪ ∅ |
| 12 | id 22 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
| 13 | unieq 4918 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅) | |
| 14 | 11, 12, 13 | 3eqtr4a 2803 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∪ 𝐴) |
| 15 | limuni 6445 | . . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
| 16 | 14, 15 | jaoi 858 | . 2 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) → 𝐴 = ∪ 𝐴) |
| 17 | 9, 16 | impbid1 225 | 1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ≠ wne 2940 ∅c0 4333 ∪ cuni 4907 Ord word 6383 Lim wlim 6385 |
| 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-11 2157 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-sn 4627 df-uni 4908 df-lim 6389 |
| This theorem is referenced by: ordzsl 7866 oeeulem 8639 cantnfp1lem2 9719 cantnflem1 9729 cnfcom2lem 9741 ordcmp 36448 onsucf1olem 43283 onov0suclim 43287 |
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