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Theorem elnn 7579
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 ordom 7578 . 2 Ord ω
2 ordtr 6198 . 2 (Ord ω → Tr ω)
3 trel 5170 . 2 (Tr ω → ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω))
41, 2, 3mp2b 10 1 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  Tr wtr 5163  Ord word 6183  ωcom 7569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-om 7570
This theorem is referenced by:  nnaordi  8233  nnmordi  8246  pssnn  8724  ssnnfi  8725  unfilem1  8770  unfilem2  8771  inf3lem5  9083  cantnflt  9123  cantnfp1lem3  9131  cantnflem1d  9139  cantnflem1  9140  cnfcomlem  9150  cnfcom  9151  infpssrlem4  9716  axdc3lem2  9861  pwfseqlem3  10070  bnj1098  31954  bnj517  32056  bnj594  32083  bnj1001  32129  bnj1118  32153  bnj1128  32159  bnj1145  32162  elhf2  33533  hfelhf  33539
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