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

Theorem onelss 4238
Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
onelss (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))

Proof of Theorem onelss
StepHypRef Expression
1 eloni 4226 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelss 4230 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
32ex 114 . 2 (Ord 𝐴 → (𝐵𝐴𝐵𝐴))
41, 3syl 14 1 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1445  wss 3013  Ord word 4213  Oncon0 4214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-in 3019  df-ss 3026  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219
This theorem is referenced by:  onelssi  4280  ssorduni  4332  onsucelsucr  4353  tfisi  4430  tfrlem9  6122  nntri2or2  6299  phpelm  6662
  Copyright terms: Public domain W3C validator