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

Theorem onelon 4087
 Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
onelon ((A On B A) → B On)

Proof of Theorem onelon
StepHypRef Expression
1 eloni 4078 . 2 (A On → Ord A)
2 ordelon 4086 . 2 ((Ord A B A) → B On)
31, 2sylan 267 1 ((A On B A) → B On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  Ord word 4065  Oncon0 4066 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071 This theorem is referenced by:  oneli  4131  ssorduni  4179  unon  4202  tfrlemibacc  5881  tfrlemibxssdm  5882  tfrlemibfn  5883  tfrexlem  5889  sucinc2  5965  oav2  5982  omv2  5984
 Copyright terms: Public domain W3C validator