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Theorem elnn 4410
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
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 3045 . . 3 (𝑦 = ∅ → (𝑦 ⊆ ω ↔ ∅ ⊆ ω))
2 sseq1 3045 . . 3 (𝑦 = 𝑥 → (𝑦 ⊆ ω ↔ 𝑥 ⊆ ω))
3 sseq1 3045 . . 3 (𝑦 = suc 𝑥 → (𝑦 ⊆ ω ↔ suc 𝑥 ⊆ ω))
4 sseq1 3045 . . 3 (𝑦 = 𝐵 → (𝑦 ⊆ ω ↔ 𝐵 ⊆ ω))
5 0ss 3318 . . 3 ∅ ⊆ ω
6 unss 3172 . . . . . 6 ((𝑥 ⊆ ω ∧ {𝑥} ⊆ ω) ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
7 vex 2622 . . . . . . . 8 𝑥 ∈ V
87snss 3561 . . . . . . 7 (𝑥 ∈ ω ↔ {𝑥} ⊆ ω)
98anbi2i 445 . . . . . 6 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω))
10 df-suc 4189 . . . . . . 7 suc 𝑥 = (𝑥 ∪ {𝑥})
1110sseq1i 3048 . . . . . 6 (suc 𝑥 ⊆ ω ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
126, 9, 113bitr4i 210 . . . . 5 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ suc 𝑥 ⊆ ω)
1312biimpi 118 . . . 4 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) → suc 𝑥 ⊆ ω)
1413expcom 114 . . 3 (𝑥 ∈ ω → (𝑥 ⊆ ω → suc 𝑥 ⊆ ω))
151, 2, 3, 4, 5, 14finds 4405 . 2 (𝐵 ∈ ω → 𝐵 ⊆ ω)
16 ssel2 3018 . . 3 ((𝐵 ⊆ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
1716ancoms 264 . 2 ((𝐴𝐵𝐵 ⊆ ω) → 𝐴 ∈ ω)
1815, 17sylan2 280 1 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1438  cun 2995  wss 2997  c0 3284  {csn 3441  suc csuc 4183  ωcom 4395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-int 3684  df-suc 4189  df-iom 4396
This theorem is referenced by:  ordom  4411  peano2b  4419  nndifsnid  6246  nnaordi  6247  nnmordi  6255  fidceq  6565  nnwetri  6606  nnti  11538  nninfsellemdc  11548  nninfsellemeq  11552  nninfsellemeqinf  11554
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