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Theorem elnn 7723
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 7721 . 2 Tr ω
2 trel 5198 . 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 5191  ωcom 7712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-lim 6271  df-om 7713
This theorem is referenced by:  nnaordi  8449  nnmordi  8462  pssnn  8951  ssnnfi  8952  ssnnfiOLD  8953  pssnnOLD  9040  unfilem1  9078  unfilem2  9079  inf3lem5  9390  cantnflt  9430  cantnfp1lem3  9438  cantnflem1d  9446  cantnflem1  9447  cnfcomlem  9457  cnfcom  9458  ttrcltr  9474  ttrclselem2  9484  infpssrlem4  10062  axdc3lem2  10207  pwfseqlem3  10416  bnj1098  32763  bnj517  32865  bnj594  32892  bnj1001  32939  bnj1118  32964  bnj1128  32970  bnj1145  32973  elhf2  34477  hfelhf  34483
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