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Theorem onelss 4073
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 4061 . 2 (A On → Ord A)
2 ordelss 4065 . . 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 1374  wss 2894  Ord word 4048  Oncon0 4049
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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054
This theorem is referenced by:  onelssi  4116  ssorduni  4163  onsucelsucr  4183  tfisi  4237  tfrlem9  5857
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