Theorem elnn 7852

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 7850	. 2 ⊢ Tr ω
2		trel 5212	. 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 399   ∈ wcel 2141  Tr wtr 5204  ωcom 7841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-ord 6344  df-on 6345  df-lim 6346  df-om 7842

This theorem is referenced by:  nnaordi  8582  nnmordi  8595  pssnn  9131  ssnnfi  9132  unfilem1  9243  unfilem2  9244  inf3lem5  9581  cantnflt  9621  cantnfp1lem3  9629  cantnflem1d  9637  cantnflem1  9638  cnfcomlem  9648  cnfcom  9649  ttrcltr  9665  ttrclselem2  9675  infpssrlem4  10257  axdc3lem2  10402  pwfseqlem3  10612  oldfi  27995  n0bday  28433  onltn0s  28439  bnj1098  35040  bnj517  35141  bnj594  35168  bnj1001  35215  bnj1118  35240  bnj1128  35246  bnj1145  35249  fineqvnttrclselem2  35379  fineqvnttrclselem3  35380  elhf2  36486  hfelhf  36492