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Mirrors > Home > ILE Home > Th. List > smodm | GIF version |
Description: The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Ref | Expression |
---|---|
smodm | ⊢ (Smo 𝐴 → Ord dom 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-smo 6284 | . 2 ⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))) | |
2 | 1 | simp2bi 1013 | 1 ⊢ (Smo 𝐴 → Ord dom 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ∀wral 2455 Ord word 4361 Oncon0 4362 dom cdm 4625 ⟶wf 5211 ‘cfv 5215 Smo wsmo 6283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-smo 6284 |
This theorem is referenced by: smores2 6292 smodm2 6293 smoel 6298 |
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