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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 3014 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
2 | df-lim 6189 | . . . . . . . . 9 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
3 | 2 | biimpri 229 | . . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴) |
4 | 3 | 3exp 1111 | . . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴))) |
5 | 1, 4 | syl5bir 244 | . . . . . 6 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴))) |
6 | 5 | com23 86 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (¬ 𝐴 = ∅ → Lim 𝐴))) |
7 | 6 | imp 407 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (¬ 𝐴 = ∅ → Lim 𝐴)) |
8 | 7 | orrd 857 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴)) |
9 | 8 | ex 413 | . 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴))) |
10 | uni0 4857 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
11 | 10 | eqcomi 2827 | . . . 4 ⊢ ∅ = ∪ ∅ |
12 | id 22 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
13 | unieq 4838 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅) | |
14 | 11, 12, 13 | 3eqtr4a 2879 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∪ 𝐴) |
15 | limuni 6244 | . . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
16 | 14, 15 | jaoi 851 | . 2 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) → 𝐴 = ∪ 𝐴) |
17 | 9, 16 | impbid1 226 | 1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ≠ wne 3013 ∅c0 4288 ∪ cuni 4830 Ord word 6183 Lim wlim 6185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-uni 4831 df-lim 6189 |
This theorem is referenced by: ordzsl 7549 oeeulem 8216 cantnfp1lem2 9130 cantnflem1 9140 cnfcom2lem 9152 ordcmp 33692 |
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