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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 4164 | . 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | |
2 | 1 | simp3bi 958 | 1 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1287 ∈ wcel 1436 ∅c0 3272 ∪ cuni 3630 Ord word 4156 Lim wlim 4158 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-ilim 4163 |
This theorem is referenced by: limuni2 4191 nlimsucg 4348 freccllem 6102 frecfcllem 6104 frecsuclem 6106 |
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