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Theorem elnn 7241
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 7240 . 2 Ord ω
2 ordtr 5898 . 2 (Ord ω → Tr ω)
3 trel 4911 . 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 383  wcel 2139  Tr wtr 4904  Ord word 5883  ωcom 7231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-om 7232
This theorem is referenced by:  nnaordi  7869  nnmordi  7882  pssnn  8345  ssnnfi  8346  unfilem1  8391  unfilem2  8392  inf3lem5  8704  cantnflt  8744  cantnfp1lem3  8752  cantnflem1d  8760  cantnflem1  8761  cnfcomlem  8771  cnfcom  8772  infpssrlem4  9340  axdc3lem2  9485  pwfseqlem3  9694  bnj1098  31182  bnj517  31283  bnj594  31310  bnj1001  31356  bnj1118  31380  bnj1128  31386  bnj1145  31389  elhf2  32609  hfelhf  32615
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