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Theorem elnn 7868
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 7866 . 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 396  wcel 2106  Tr wtr 5265  ωcom 7857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-lim 6369  df-om 7858
This theorem is referenced by:  nnaordi  8620  nnmordi  8633  pssnn  9170  ssnnfi  9171  ssnnfiOLD  9172  pssnnOLD  9267  unfilem1  9312  unfilem2  9313  inf3lem5  9629  cantnflt  9669  cantnfp1lem3  9677  cantnflem1d  9685  cantnflem1  9686  cnfcomlem  9696  cnfcom  9697  ttrcltr  9713  ttrclselem2  9723  infpssrlem4  10303  axdc3lem2  10448  pwfseqlem3  10657  bnj1098  33863  bnj517  33965  bnj594  33992  bnj1001  34039  bnj1118  34064  bnj1128  34070  bnj1145  34073  elhf2  35222  hfelhf  35228
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