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Theorem onelss 4090
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  On 
C_

Proof of Theorem onelss
StepHypRef Expression
1 eloni 4078 . 2  On  Ord
2 ordelss 4082 . . 3  Ord  C_
32ex 108 . 2  Ord  C_
41, 3syl 14 1  On 
C_
Colors of variables: wff set class
Syntax hints:   wi 4   wcel 1390    C_ wss 2911   Ord word 4065   Oncon0 4066
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071
This theorem is referenced by:  onelssi  4132  ssorduni  4179  onsucelsucr  4199  tfisi  4253  tfrlem9  5876
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