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Theorem elnn 7814
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 7812 . 2 Tr ω
2 trel 5232 . 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 397  wcel 2107  Tr wtr 5223  ωcom 7803
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-on 6322  df-lim 6323  df-om 7804
This theorem is referenced by:  nnaordi  8566  nnmordi  8579  pssnn  9115  ssnnfi  9116  ssnnfiOLD  9117  pssnnOLD  9212  unfilem1  9257  unfilem2  9258  inf3lem5  9573  cantnflt  9613  cantnfp1lem3  9621  cantnflem1d  9629  cantnflem1  9630  cnfcomlem  9640  cnfcom  9641  ttrcltr  9657  ttrclselem2  9667  infpssrlem4  10247  axdc3lem2  10392  pwfseqlem3  10601  bnj1098  33452  bnj517  33554  bnj594  33581  bnj1001  33628  bnj1118  33653  bnj1128  33659  bnj1145  33662  elhf2  34806  hfelhf  34812
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