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Theorem el1o 8110
 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 8102 . . 3 1o = {∅}
21eleq2i 2881 . 2 (𝐴 ∈ 1o𝐴 ∈ {∅})
3 0ex 5176 . . 3 ∅ ∈ V
43elsn2 4564 . 2 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
52, 4bitri 278 1 (𝐴 ∈ 1o𝐴 = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∅c0 4243  {csn 4525  1oc1o 8081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-nul 5175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-suc 6166  df-1o 8088 This theorem is referenced by:  0lt1o  8115  oelim2  8207  oeeulem  8213  oaabs2  8258  cantnff  9124  cnfcom3lem  9153  cfsuc  9671  pf1ind  20989  mavmul0  21167  cramer0  21305
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