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Theorem elnn 4255
 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
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 2943 . . 3 (y = ∅ → (y ⊆ 𝜔 ↔ ∅ ⊆ 𝜔))
2 sseq1 2943 . . 3 (y = x → (y ⊆ 𝜔 ↔ x ⊆ 𝜔))
3 sseq1 2943 . . 3 (y = suc x → (y ⊆ 𝜔 ↔ suc x ⊆ 𝜔))
4 sseq1 2943 . . 3 (y = B → (y ⊆ 𝜔 ↔ B ⊆ 𝜔))
5 0ss 3232 . . 3 ∅ ⊆ 𝜔
6 unss 3094 . . . . . 6 ((x ⊆ 𝜔 {x} ⊆ 𝜔) ↔ (x ∪ {x}) ⊆ 𝜔)
7 vex 2538 . . . . . . . 8 x V
87snss 3468 . . . . . . 7 (x 𝜔 ↔ {x} ⊆ 𝜔)
98anbi2i 433 . . . . . 6 ((x ⊆ 𝜔 x 𝜔) ↔ (x ⊆ 𝜔 {x} ⊆ 𝜔))
10 df-suc 4057 . . . . . . 7 suc x = (x ∪ {x})
1110sseq1i 2946 . . . . . 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 4250 . 2 (B 𝜔 → B ⊆ 𝜔)
16 ssel2 2917 . . 3 ((B ⊆ 𝜔 A B) → A 𝜔)
1716ancoms 255 . 2 ((A B B ⊆ 𝜔) → A 𝜔)
1815, 17sylan2 270 1 ((A B B 𝜔) → A 𝜔)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1374   ∪ cun 2892   ⊆ wss 2894  ∅c0 3201  {csn 3350  suc csuc 4051  𝜔com 4240 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-iinf 4238 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241 This theorem is referenced by:  ordom  4256  peano2b  4264  nnaordi  5992  nnmordi  6000
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