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Theorem el1o 6346
 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 6338 . . 3 1o = {∅}
21eleq2i 2208 . 2 (𝐴 ∈ 1o𝐴 ∈ {∅})
3 0ex 4065 . . 3 ∅ ∈ V
43elsn2 3568 . 2 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
52, 4bitri 183 1 (𝐴 ∈ 1o𝐴 = ∅)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∅c0 3370  {csn 3534  1oc1o 6318 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-nul 4064 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1738  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-v 2693  df-dif 3080  df-un 3082  df-nul 3371  df-sn 3540  df-suc 4304  df-1o 6325 This theorem is referenced by:  0lt1o  6349  map0e  6592  map1  6718  omp1eomlem  6996  ctmlemr  7010  ctssdclemn0  7012  exmidfodomrlemeldju  7084  exmidfodomrlemreseldju  7085  pw1on  7106  1tonninf  10273
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