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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 4287 | . 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | |
2 | 1 | simp3bi 998 | 1 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∅c0 3358 ∪ cuni 3731 Ord word 4279 Lim wlim 4281 |
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 964 df-ilim 4286 |
This theorem is referenced by: limuni2 4314 nlimsucg 4476 freccllem 6292 frecfcllem 6294 frecsuclem 6296 |
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