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Theorem elnn 7023
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 7022 . 2 Ord ω
2 ordtr 5699 . 2 (Ord ω → Tr ω)
3 trel 4724 . 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 384  wcel 1992  Tr wtr 4717  Ord word 5684  ωcom 7013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-om 7014
This theorem is referenced by:  nnaordi  7644  nnmordi  7657  pssnn  8123  ssnnfi  8124  unfilem1  8169  unfilem2  8170  inf3lem5  8474  cantnflt  8514  cantnfp1lem3  8522  cantnflem1d  8530  cantnflem1  8531  cnfcomlem  8541  cnfcom  8542  infpssrlem4  9073  axdc3lem2  9218  pwfseqlem3  9427  bnj1098  30554  bnj517  30655  bnj594  30682  bnj1001  30728  bnj1118  30752  bnj1128  30758  bnj1145  30761  elhf2  31916  hfelhf  31922
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