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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 2933 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
2 | df-lim 6360 | . . . . . . . . 9 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
3 | 2 | biimpri 227 | . . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴) |
4 | 3 | 3exp 1116 | . . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴))) |
5 | 1, 4 | biimtrrid 242 | . . . . . 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 860 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴)) |
9 | 8 | ex 412 | . 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴))) |
10 | uni0 4930 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
11 | 10 | eqcomi 2733 | . . . 4 ⊢ ∅ = ∪ ∅ |
12 | id 22 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
13 | unieq 4911 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅) | |
14 | 11, 12, 13 | 3eqtr4a 2790 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∪ 𝐴) |
15 | limuni 6416 | . . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
16 | 14, 15 | jaoi 854 | . 2 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) → 𝐴 = ∪ 𝐴) |
17 | 9, 16 | impbid1 224 | 1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ≠ wne 2932 ∅c0 4315 ∪ cuni 4900 Ord word 6354 Lim wlim 6356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-v 3468 df-dif 3944 df-in 3948 df-ss 3958 df-nul 4316 df-sn 4622 df-uni 4901 df-lim 6360 |
This theorem is referenced by: ordzsl 7828 oeeulem 8597 cantnfp1lem2 9671 cantnflem1 9681 cnfcom2lem 9693 ordcmp 35823 onsucf1olem 42534 onov0suclim 42538 |
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