ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elnn Structured version   GIF version

Theorem elnn 4271
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 2960 . . 3 (y = ∅ → (y ⊆ 𝜔 ↔ ∅ ⊆ 𝜔))
2 sseq1 2960 . . 3 (y = x → (y ⊆ 𝜔 ↔ x ⊆ 𝜔))
3 sseq1 2960 . . 3 (y = suc x → (y ⊆ 𝜔 ↔ suc x ⊆ 𝜔))
4 sseq1 2960 . . 3 (y = B → (y ⊆ 𝜔 ↔ B ⊆ 𝜔))
5 0ss 3249 . . 3 ∅ ⊆ 𝜔
6 unss 3111 . . . . . 6 ((x ⊆ 𝜔 {x} ⊆ 𝜔) ↔ (x ∪ {x}) ⊆ 𝜔)
7 vex 2554 . . . . . . . 8 x V
87snss 3485 . . . . . . 7 (x 𝜔 ↔ {x} ⊆ 𝜔)
98anbi2i 430 . . . . . 6 ((x ⊆ 𝜔 x 𝜔) ↔ (x ⊆ 𝜔 {x} ⊆ 𝜔))
10 df-suc 4074 . . . . . . 7 suc x = (x ∪ {x})
1110sseq1i 2963 . . . . . 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 4266 . 2 (B 𝜔 → B ⊆ 𝜔)
16 ssel2 2934 . . 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 1390  cun 2909  wss 2911  c0 3218  {csn 3367  suc csuc 4068  𝜔com 4256
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257
This theorem is referenced by:  ordom  4272  peano2b  4280  nnaordi  6017  nnmordi  6025
  Copyright terms: Public domain W3C validator