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Theorem el1o 6157
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 6150 . . 3 1𝑜 = {∅}
21eleq2i 2151 . 2 (𝐴 ∈ 1𝑜𝐴 ∈ {∅})
3 0ex 3943 . . 3 ∅ ∈ V
43elsn2 3463 . 2 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
52, 4bitri 182 1 (𝐴 ∈ 1𝑜𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1287  wcel 1436  c0 3275  {csn 3431  1𝑜c1o 6130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-nul 3942
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-dif 2990  df-un 2992  df-nul 3276  df-sn 3437  df-suc 4174  df-1o 6137
This theorem is referenced by:  0lt1o  6160  map0e  6397  map1  6483  exmidfodomrlemeldju  6772  exmidfodomrlemreseldju  6773  1tonninf  9777
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