Theorem elnn 7872

Description: A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)

Assertion

Ref	Expression
elnn	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)

Proof of Theorem elnn

Step	Hyp	Ref	Expression
1		trom 7870	. 2 ⊢ Tr ω
2		trel 5238	. 2 ⊢ (Tr ω → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Tr wtr 5229  ωcom 7861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-tr 5230  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-ord 6355  df-on 6356  df-lim 6357  df-om 7862

This theorem is referenced by:  nnaordi  8630  nnmordi  8643  pssnn  9182  ssnnfi  9183  unfilem1  9315  unfilem2  9316  inf3lem5  9646  cantnflt  9686  cantnfp1lem3  9694  cantnflem1d  9702  cantnflem1  9703  cnfcomlem  9713  cnfcom  9714  ttrcltr  9730  ttrclselem2  9740  infpssrlem4  10320  axdc3lem2  10465  pwfseqlem3  10674  oldfi  27877  n0sbday  28296  onltn0s  28300  zs12bday  28395  bnj1098  34814  bnj517  34916  bnj594  34943  bnj1001  34990  bnj1118  35015  bnj1128  35021  bnj1145  35024  elhf2  36193  hfelhf  36199