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Theorem el1o 6020
 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 6013 . . 3 1𝑜 = {∅}
21eleq2i 2104 . 2 (𝐴 ∈ 1𝑜𝐴 ∈ {∅})
3 0ex 3884 . . 3 ∅ ∈ V
43elsn2 3405 . 2 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
52, 4bitri 173 1 (𝐴 ∈ 1𝑜𝐴 = ∅)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∅c0 3224  {csn 3375  1𝑜c1o 5994 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-nul 3225  df-sn 3381  df-suc 4108  df-1o 6001 This theorem is referenced by:  0lt1o  6023
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