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Theorem elnn 7862
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 7860 . 2 Tr ω
2 trel 5273 . 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 5264  ωcom 7851
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 5298  ax-nul 5305  ax-pr 5426
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 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-lim 6366  df-om 7852
This theorem is referenced by:  nnaordi  8614  nnmordi  8627  pssnn  9164  ssnnfi  9165  ssnnfiOLD  9166  pssnnOLD  9261  unfilem1  9306  unfilem2  9307  inf3lem5  9623  cantnflt  9663  cantnfp1lem3  9671  cantnflem1d  9679  cantnflem1  9680  cnfcomlem  9690  cnfcom  9691  ttrcltr  9707  ttrclselem2  9717  infpssrlem4  10297  axdc3lem2  10442  pwfseqlem3  10651  bnj1098  33782  bnj517  33884  bnj594  33911  bnj1001  33958  bnj1118  33983  bnj1128  33989  bnj1145  33992  elhf2  35135  hfelhf  35141
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