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Theorem el1o 8118
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 8110 . . 3 1o = {∅}
21eleq2i 2908 . 2 (𝐴 ∈ 1o𝐴 ∈ {∅})
3 0ex 5207 . . 3 ∅ ∈ V
43elsn2 4600 . 2 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
52, 4bitri 276 1 (𝐴 ∈ 1o𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1530  wcel 2107  c0 4294  {csn 4563  1oc1o 8089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-nul 5206
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-v 3501  df-dif 3942  df-un 3944  df-nul 4295  df-sn 4564  df-suc 6194  df-1o 8096
This theorem is referenced by:  0lt1o  8123  oelim2  8214  oeeulem  8220  oaabs2  8265  cantnff  9129  cnfcom3lem  9158  cfsuc  9671  pf1ind  20436  mavmul0  21079  cramer0  21217
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