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Mirrors > Home > ILE Home > Th. List > limord | GIF version |
Description: A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.) |
Ref | Expression |
---|---|
limord | ⊢ (Lim 𝐴 → Ord 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dflim2 4332 | . 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | |
2 | 1 | simp1bi 997 | 1 ⊢ (Lim 𝐴 → Ord 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∈ wcel 2128 ∅c0 3395 ∪ cuni 3774 Ord word 4324 Lim wlim 4326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-ilim 4331 |
This theorem is referenced by: limelon 4361 nlimsucg 4527 |
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