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Theorem onelss 4071
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 → (B ABA))

Proof of Theorem onelss
StepHypRef Expression
1 eloni 4059 . 2 (A On → Ord A)
2 ordelss 4063 . . 3 ((Ord A B A) → BA)
32ex 108 . 2 (Ord A → (B ABA))
41, 3syl 14 1 (A On → (B ABA))
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1375  wss 2893  Ord word 4046  Oncon0 4047
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-in 2900  df-ss 2907  df-uni 3554  df-tr 3828  df-iord 4050  df-on 4052
This theorem is referenced by:  onelssi  4114  ssorduni  4161  onsucelsucr  4181  tfisi  4235  tfrlem9  5852
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