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

Theorem onss 4409
Description: An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
onss (𝐴 ∈ On → 𝐴 ⊆ On)

Proof of Theorem onss
StepHypRef Expression
1 eloni 4297 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordsson 4408 . 2 (Ord 𝐴𝐴 ⊆ On)
31, 2syl 14 1 (𝐴 ∈ On → 𝐴 ⊆ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1480  wss 3071  Ord word 4284  Oncon0 4285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290
This theorem is referenced by:  onuni  4410  onssi  4431  tfrexlem  6231  tfri3  6264  rdgivallem  6278  bj-omssonALT  13191
  Copyright terms: Public domain W3C validator