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Theorem onelss 6428
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 6396 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelss 6402 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
32ex 412 . 2 (Ord 𝐴 → (𝐵𝐴𝐵𝐴))
41, 3syl 17 1 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wss 3963  Ord word 6385  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-ss 3980  df-uni 4913  df-tr 5266  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  ordunidif  6435  onelssi  6501  ssorduni  7798  sucexeloniOLD  7830  suceloniOLD  7832  tfisi  7880  poseq  8182  tfrlem9  8424  tfrlem11  8427  oaordex  8595  oaass  8598  odi  8616  omass  8617  oewordri  8629  nnaordex  8675  domtriord  9162  hartogs  9582  card2on  9592  tskwe  9988  infxpenlem  10051  cfub  10287  cfsuc  10295  coflim  10299  hsmexlem2  10465  ondomon  10601  pwcfsdom  10621  inar1  10813  tskord  10818  grudomon  10855  gruina  10856  sltres  27722  nosupno  27763  nosupbday  27765  noinfno  27778  oldssmade  27931  madebday  27953  mulsproplem13  28169  mulsproplem14  28170  dfrdg2  35777  aomclem6  43048  nnoeomeqom  43302  naddgeoa  43384  naddwordnexlem1  43387  naddwordnexlem4  43391  iscard5  43526
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