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Theorem elnn 7898
Description: A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
Assertion
Ref Expression
elnn ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)

Proof of Theorem elnn
StepHypRef Expression
1 trom 7896 . 2 Tr ω
2 trel 5274 . 2 (Tr ω → ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω))
31, 2ax-mp 5 1 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  Tr wtr 5265  ωcom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-om 7888
This theorem is referenced by:  nnaordi  8655  nnmordi  8668  pssnn  9207  ssnnfi  9208  unfilem1  9341  unfilem2  9342  inf3lem5  9670  cantnflt  9710  cantnfp1lem3  9718  cantnflem1d  9726  cantnflem1  9727  cnfcomlem  9737  cnfcom  9738  ttrcltr  9754  ttrclselem2  9764  infpssrlem4  10344  axdc3lem2  10489  pwfseqlem3  10698  oldfi  27966  n0sbday  28369  pw2bday  28433  zs12bday  28439  bnj1098  34776  bnj517  34878  bnj594  34905  bnj1001  34952  bnj1118  34977  bnj1128  34983  bnj1145  34986  elhf2  36157  hfelhf  36163
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