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Theorem el1o 7845
 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 7838 . . 3 1o = {∅}
21eleq2i 2897 . 2 (𝐴 ∈ 1o𝐴 ∈ {∅})
3 0ex 5013 . . 3 ∅ ∈ V
43elsn2 4431 . 2 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
52, 4bitri 267 1 (𝐴 ∈ 1o𝐴 = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1658   ∈ wcel 2166  ∅c0 4143  {csn 4396  1oc1o 7818 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-nul 5012 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-v 3415  df-dif 3800  df-un 3802  df-nul 4144  df-sn 4397  df-suc 5968  df-1o 7825 This theorem is referenced by:  0lt1o  7850  oelim2  7941  oeeulem  7947  oaabs2  7991  map0eOLD  8160  cantnff  8847  cnfcom3lem  8876  cfsuc  9393  pf1ind  20078  mavmul0  20725  cramer0  20865
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