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| Mirrors > Home > MPE Home > Th. List > Mathboxes > limiun | Structured version Visualization version GIF version | ||
| Description: A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6445. (Contributed by RP, 27-Jan-2025.) |
| Ref | Expression |
|---|---|
| limiun | ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | limuni 6445 | . 2 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
| 2 | uniiun 5058 | . 2 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
| 3 | 1, 2 | eqtrdi 2793 | 1 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∪ cuni 4907 ∪ ciun 4991 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-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-rex 3071 df-uni 4908 df-iun 4993 df-lim 6389 |
| This theorem is referenced by: (None) |
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