ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  smodm2 Structured version   GIF version

Theorem smodm2 5832
Description: The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smodm2 ((𝐹 Fn A Smo 𝐹) → Ord A)

Proof of Theorem smodm2
StepHypRef Expression
1 smodm 5828 . 2 (Smo 𝐹 → Ord dom 𝐹)
2 fndm 4924 . . . 4 (𝐹 Fn A → dom 𝐹 = A)
3 ordeq 4058 . . . 4 (dom 𝐹 = A → (Ord dom 𝐹 ↔ Ord A))
42, 3syl 14 . . 3 (𝐹 Fn A → (Ord dom 𝐹 ↔ Ord A))
54biimpa 280 . 2 ((𝐹 Fn A Ord dom 𝐹) → Ord A)
61, 5sylan2 270 1 ((𝐹 Fn A Smo 𝐹) → Ord A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  Ord word 4048  dom cdm 4272   Fn wfn 4824  Smo wsmo 5822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-in 2901  df-ss 2908  df-uni 3555  df-tr 3829  df-iord 4052  df-fn 4832  df-smo 5823
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator