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Theorem onelss 4177
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 4165 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordelss 4169 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
32ex 113 . 2 (Ord 𝐴 → (𝐵𝐴𝐵𝐴))
41, 3syl 14 1 (𝐴 ∈ On → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  wss 2984  Ord word 4152  Oncon0 4153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-in 2990  df-ss 2997  df-uni 3628  df-tr 3902  df-iord 4156  df-on 4158
This theorem is referenced by:  onelssi  4219  ssorduni  4266  onsucelsucr  4287  tfisi  4364  tfrlem9  6015  nntri2or2  6190  phpelm  6511
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