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Theorem smodm 8347
Description: The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
Assertion
Ref Expression
smodm (Smo 𝐴 β†’ Ord dom 𝐴)

Proof of Theorem smodm
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-smo 8342 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟢On ∧ Ord dom 𝐴 ∧ βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦))))
21simp2bi 1146 1 (Smo 𝐴 β†’ Ord dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2106  βˆ€wral 3061  dom cdm 5675  Ord word 6360  Oncon0 6361  βŸΆwf 6536  β€˜cfv 6540  Smo wsmo 8341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-smo 8342
This theorem is referenced by:  smores2  8350  smodm2  8351  smoel  8356
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