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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 5924 | . 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 954 | 1 ⊢ (Smo 𝐴 → Ord dom 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 ∀wral 2348 Ord word 4117 Oncon0 4118 dom cdm 4363 ⟶wf 4918 ‘cfv 4922 Smo wsmo 5923 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-smo 5924 |
This theorem is referenced by: smores2 5932 smodm2 5933 smoel 5938 |
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