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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 2795 | . . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
2 | df-lim 5728 | . . . . . . . . 9 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
3 | 2 | biimpri 218 | . . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) → Lim 𝐴) |
4 | 3 | 3exp 1264 | . . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴))) |
5 | 1, 4 | syl5bir 233 | . . . . . 6 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∅ → (𝐴 = ∪ 𝐴 → Lim 𝐴))) |
6 | 5 | com23 86 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (¬ 𝐴 = ∅ → Lim 𝐴))) |
7 | 6 | imp 445 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (¬ 𝐴 = ∅ → Lim 𝐴)) |
8 | 7 | orrd 393 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴)) |
9 | 8 | ex 450 | . 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴))) |
10 | uni0 4465 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
11 | 10 | eqcomi 2631 | . . . 4 ⊢ ∅ = ∪ ∅ |
12 | id 22 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
13 | unieq 4444 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅) | |
14 | 11, 12, 13 | 3eqtr4a 2682 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∪ 𝐴) |
15 | limuni 5785 | . . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
16 | 14, 15 | jaoi 394 | . 2 ⊢ ((𝐴 = ∅ ∨ Lim 𝐴) → 𝐴 = ∪ 𝐴) |
17 | 9, 16 | impbid1 215 | 1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ≠ wne 2794 ∅c0 3915 ∪ cuni 4436 Ord word 5722 Lim wlim 5724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-uni 4437 df-lim 5728 |
This theorem is referenced by: ordzsl 7045 oeeulem 7681 cantnfp1lem2 8576 cantnflem1 8586 cnfcom2lem 8598 ordcmp 32446 |
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