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Theorem onelss 4047
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 4035 . 2 (A On → Ord A)
2 ordelss 4039 . . 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 2895  Ord word 4023  Oncon0 4024
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-in 2902  df-ss 2909  df-uni 3533  df-tr 3807  df-iord 4027  df-on 4028
This theorem is referenced by:  onelssi  4089  ssorduni  4136  onsucelsucr  4156  tfisi  4206  tfrlem9  5822
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