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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 6196 | . . . . . . . . 9 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
3 | 2 | biimpri 231 | . . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴) |
4 | 3 | 3exp 1121 | . . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴))) |
5 | 1, 4 | syl5bir 246 | . . . . . 6 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴))) |
6 | 5 | com23 86 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (¬ 𝐴 = ∅ → Lim 𝐴))) |
7 | 6 | imp 410 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (¬ 𝐴 = ∅ → Lim 𝐴)) |
8 | 7 | orrd 863 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴)) |
9 | 8 | ex 416 | . 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴))) |
10 | uni0 4835 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
11 | 10 | eqcomi 2745 | . . . 4 ⊢ ∅ = ∪ ∅ |
12 | id 22 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
13 | unieq 4816 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅) | |
14 | 11, 12, 13 | 3eqtr4a 2797 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∪ 𝐴) |
15 | limuni 6251 | . . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
16 | 14, 15 | jaoi 857 | . 2 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) → 𝐴 = ∪ 𝐴) |
17 | 9, 16 | impbid1 228 | 1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 ∧ w3a 1089 = wceq 1543 ≠ wne 2932 ∅c0 4223 ∪ cuni 4805 Ord word 6190 Lim wlim 6192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-v 3400 df-dif 3856 df-in 3860 df-ss 3870 df-nul 4224 df-sn 4528 df-uni 4806 df-lim 6196 |
This theorem is referenced by: ordzsl 7602 oeeulem 8307 cantnfp1lem2 9272 cantnflem1 9282 cnfcom2lem 9294 ordcmp 34322 |
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