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Theorem elnn 4487
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 3088 . . 3 (𝑦 = ∅ → (𝑦 ⊆ ω ↔ ∅ ⊆ ω))
2 sseq1 3088 . . 3 (𝑦 = 𝑥 → (𝑦 ⊆ ω ↔ 𝑥 ⊆ ω))
3 sseq1 3088 . . 3 (𝑦 = suc 𝑥 → (𝑦 ⊆ ω ↔ suc 𝑥 ⊆ ω))
4 sseq1 3088 . . 3 (𝑦 = 𝐵 → (𝑦 ⊆ ω ↔ 𝐵 ⊆ ω))
5 0ss 3369 . . 3 ∅ ⊆ ω
6 unss 3218 . . . . . 6 ((𝑥 ⊆ ω ∧ {𝑥} ⊆ ω) ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
7 vex 2661 . . . . . . . 8 𝑥 ∈ V
87snss 3617 . . . . . . 7 (𝑥 ∈ ω ↔ {𝑥} ⊆ ω)
98anbi2i 450 . . . . . 6 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ (𝑥 ⊆ ω ∧ {𝑥} ⊆ ω))
10 df-suc 4261 . . . . . . 7 suc 𝑥 = (𝑥 ∪ {𝑥})
1110sseq1i 3091 . . . . . 6 (suc 𝑥 ⊆ ω ↔ (𝑥 ∪ {𝑥}) ⊆ ω)
126, 9, 113bitr4i 211 . . . . 5 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) ↔ suc 𝑥 ⊆ ω)
1312biimpi 119 . . . 4 ((𝑥 ⊆ ω ∧ 𝑥 ∈ ω) → suc 𝑥 ⊆ ω)
1413expcom 115 . . 3 (𝑥 ∈ ω → (𝑥 ⊆ ω → suc 𝑥 ⊆ ω))
151, 2, 3, 4, 5, 14finds 4482 . 2 (𝐵 ∈ ω → 𝐵 ⊆ ω)
16 ssel2 3060 . . 3 ((𝐵 ⊆ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
1716ancoms 266 . 2 ((𝐴𝐵𝐵 ⊆ ω) → 𝐴 ∈ ω)
1815, 17sylan2 282 1 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1463  cun 3037  wss 3039  c0 3331  {csn 3495  suc csuc 4255  ωcom 4472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-suc 4261  df-iom 4473
This theorem is referenced by:  ordom  4488  peano2b  4496  nntr2  6365  nndifsnid  6369  nnaordi  6370  nnmordi  6378  fidceq  6729  nnwetri  6770  enumctlemm  6965  ennnfonelemdm  11839  ennnfonelemnn0  11841  nnti  13025  nninfsellemdc  13040  nninfsellemeq  13044  nninfsellemeqinf  13046
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