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Theorem elnn 7581
 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 7580 . 2 Ord ω
2 ordtr 6202 . 2 (Ord ω → Tr ω)
3 trel 5175 . 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 2107  Tr wtr 5168  Ord word 6187  ωcom 7571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-tr 5169  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-om 7572 This theorem is referenced by:  nnaordi  8237  nnmordi  8250  pssnn  8728  ssnnfi  8729  unfilem1  8774  unfilem2  8775  inf3lem5  9087  cantnflt  9127  cantnfp1lem3  9135  cantnflem1d  9143  cantnflem1  9144  cnfcomlem  9154  cnfcom  9155  infpssrlem4  9720  axdc3lem2  9865  pwfseqlem3  10074  bnj1098  31943  bnj517  32045  bnj594  32072  bnj1001  32118  bnj1118  32142  bnj1128  32148  bnj1145  32151  elhf2  33522  hfelhf  33528
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