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Theorem elnn 4355
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 2993 . . 3 (𝑦 = ∅ → (𝑦 ⊆ ω ↔ ∅ ⊆ ω))
2 sseq1 2993 . . 3 (𝑦 = 𝑥 → (𝑦 ⊆ ω ↔ 𝑥 ⊆ ω))
3 sseq1 2993 . . 3 (𝑦 = suc 𝑥 → (𝑦 ⊆ ω ↔ suc 𝑥 ⊆ ω))
4 sseq1 2993 . . 3 (𝑦 = 𝐵 → (𝑦 ⊆ ω ↔ 𝐵 ⊆ ω))
5 0ss 3282 . . 3 ∅ ⊆ ω
6 unss 3144 . . . . . 6 ((𝑥 ⊆ ω ∧ {𝑥} ⊆ ω) ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
7 vex 2577 . . . . . . . 8 𝑥 ∈ V
87snss 3521 . . . . . . 7 (𝑥 ∈ ω ↔ {𝑥} ⊆ ω)
98anbi2i 438 . . . . . 6 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω))
10 df-suc 4135 . . . . . . 7 suc 𝑥 = (𝑥 ∪ {𝑥})
1110sseq1i 2996 . . . . . 6 (suc 𝑥 ⊆ ω ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
126, 9, 113bitr4i 205 . . . . 5 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ suc 𝑥 ⊆ ω)
1312biimpi 117 . . . 4 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) → suc 𝑥 ⊆ ω)
1413expcom 113 . . 3 (𝑥 ∈ ω → (𝑥 ⊆ ω → suc 𝑥 ⊆ ω))
151, 2, 3, 4, 5, 14finds 4350 . 2 (𝐵 ∈ ω → 𝐵 ⊆ ω)
16 ssel2 2967 . . 3 ((𝐵 ⊆ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
1716ancoms 259 . 2 ((𝐴𝐵𝐵 ⊆ ω) → 𝐴 ∈ ω)
1815, 17sylan2 274 1 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  cun 2942  wss 2944  c0 3251  {csn 3402  suc csuc 4129  ωcom 4340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-uni 3608  df-int 3643  df-suc 4135  df-iom 4341
This theorem is referenced by:  ordom  4356  peano2b  4364  nndifsnid  6110  nnaordi  6111  nnmordi  6119  fidceq  6360  nnwetri  6384
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