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Theorem elnn 4328
 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 2966 . . 3 (𝑦 = ∅ → (𝑦 ⊆ ω ↔ ∅ ⊆ ω))
2 sseq1 2966 . . 3 (𝑦 = 𝑥 → (𝑦 ⊆ ω ↔ 𝑥 ⊆ ω))
3 sseq1 2966 . . 3 (𝑦 = suc 𝑥 → (𝑦 ⊆ ω ↔ suc 𝑥 ⊆ ω))
4 sseq1 2966 . . 3 (𝑦 = 𝐵 → (𝑦 ⊆ ω ↔ 𝐵 ⊆ ω))
5 0ss 3255 . . 3 ∅ ⊆ ω
6 unss 3117 . . . . . 6 ((𝑥 ⊆ ω ∧ {𝑥} ⊆ ω) ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
7 vex 2560 . . . . . . . 8 𝑥 ∈ V
87snss 3494 . . . . . . 7 (𝑥 ∈ ω ↔ {𝑥} ⊆ ω)
98anbi2i 430 . . . . . 6 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω))
10 df-suc 4108 . . . . . . 7 suc 𝑥 = (𝑥 ∪ {𝑥})
1110sseq1i 2969 . . . . . 6 (suc 𝑥 ⊆ ω ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
126, 9, 113bitr4i 201 . . . . 5 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ suc 𝑥 ⊆ ω)
1312biimpi 113 . . . 4 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) → suc 𝑥 ⊆ ω)
1413expcom 109 . . 3 (𝑥 ∈ ω → (𝑥 ⊆ ω → suc 𝑥 ⊆ ω))
151, 2, 3, 4, 5, 14finds 4323 . 2 (𝐵 ∈ ω → 𝐵 ⊆ ω)
16 ssel2 2940 . . 3 ((𝐵 ⊆ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
1716ancoms 255 . 2 ((𝐴𝐵𝐵 ⊆ ω) → 𝐴 ∈ ω)
1815, 17sylan2 270 1 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393   ∪ cun 2915   ⊆ wss 2917  ∅c0 3224  {csn 3375  suc csuc 4102  ωcom 4313 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314 This theorem is referenced by:  ordom  4329  peano2b  4337  nndifsnid  6080  nnaordi  6081  nnmordi  6089  fidceq  6330  nnwetri  6354
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