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Mirrors > Home > ILE Home > Th. List > limuni | GIF version |
Description: A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.) |
Ref | Expression |
---|---|
limuni | ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dflim2 4353 | . 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | |
2 | 1 | simp3bi 1009 | 1 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 ∅c0 3414 ∪ cuni 3794 Ord word 4345 Lim wlim 4347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-ilim 4352 |
This theorem is referenced by: limuni2 4380 nlimsucg 4548 freccllem 6378 frecfcllem 6380 frecsuclem 6382 |
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