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Theorem el1o 8113
Description: Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
Assertion
Ref Expression
el1o (𝐴 ∈ 1o𝐴 = ∅)

Proof of Theorem el1o
StepHypRef Expression
1 df1o2 8105 . . 3 1o = {∅}
21eleq2i 2901 . 2 (𝐴 ∈ 1o𝐴 ∈ {∅})
3 0ex 5202 . . 3 ∅ ∈ V
43elsn2 4594 . 2 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
52, 4bitri 276 1 (𝐴 ∈ 1o𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  c0 4288  {csn 4557  1oc1o 8084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-dif 3936  df-un 3938  df-nul 4289  df-sn 4558  df-suc 6190  df-1o 8091
This theorem is referenced by:  0lt1o  8118  oelim2  8210  oeeulem  8216  oaabs2  8261  cantnff  9125  cnfcom3lem  9154  cfsuc  9667  pf1ind  20446  mavmul0  21089  cramer0  21227
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