ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nn0suc Structured version   GIF version

Theorem nn0suc 4270
Description: A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
nn0suc (A 𝜔 → (A = ∅ x 𝜔 A = suc x))
Distinct variable group:   x,A

Proof of Theorem nn0suc
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2043 . . 3 (y = ∅ → (y = ∅ ↔ ∅ = ∅))
2 eqeq1 2043 . . . 4 (y = ∅ → (y = suc x ↔ ∅ = suc x))
32rexbidv 2321 . . 3 (y = ∅ → (x 𝜔 y = suc xx 𝜔 ∅ = suc x))
41, 3orbi12d 706 . 2 (y = ∅ → ((y = ∅ x 𝜔 y = suc x) ↔ (∅ = ∅ x 𝜔 ∅ = suc x)))
5 eqeq1 2043 . . 3 (y = z → (y = ∅ ↔ z = ∅))
6 eqeq1 2043 . . . 4 (y = z → (y = suc xz = suc x))
76rexbidv 2321 . . 3 (y = z → (x 𝜔 y = suc xx 𝜔 z = suc x))
85, 7orbi12d 706 . 2 (y = z → ((y = ∅ x 𝜔 y = suc x) ↔ (z = ∅ x 𝜔 z = suc x)))
9 eqeq1 2043 . . 3 (y = suc z → (y = ∅ ↔ suc z = ∅))
10 eqeq1 2043 . . . 4 (y = suc z → (y = suc x ↔ suc z = suc x))
1110rexbidv 2321 . . 3 (y = suc z → (x 𝜔 y = suc xx 𝜔 suc z = suc x))
129, 11orbi12d 706 . 2 (y = suc z → ((y = ∅ x 𝜔 y = suc x) ↔ (suc z = ∅ x 𝜔 suc z = suc x)))
13 eqeq1 2043 . . 3 (y = A → (y = ∅ ↔ A = ∅))
14 eqeq1 2043 . . . 4 (y = A → (y = suc xA = suc x))
1514rexbidv 2321 . . 3 (y = A → (x 𝜔 y = suc xx 𝜔 A = suc x))
1613, 15orbi12d 706 . 2 (y = A → ((y = ∅ x 𝜔 y = suc x) ↔ (A = ∅ x 𝜔 A = suc x)))
17 eqid 2037 . . 3 ∅ = ∅
1817orci 649 . 2 (∅ = ∅ x 𝜔 ∅ = suc x)
19 eqid 2037 . . . . 5 suc z = suc z
20 suceq 4105 . . . . . . 7 (x = z → suc x = suc z)
2120eqeq2d 2048 . . . . . 6 (x = z → (suc z = suc x ↔ suc z = suc z))
2221rspcev 2650 . . . . 5 ((z 𝜔 suc z = suc z) → x 𝜔 suc z = suc x)
2319, 22mpan2 401 . . . 4 (z 𝜔 → x 𝜔 suc z = suc x)
2423olcd 652 . . 3 (z 𝜔 → (suc z = ∅ x 𝜔 suc z = suc x))
2524a1d 22 . 2 (z 𝜔 → ((z = ∅ x 𝜔 z = suc x) → (suc z = ∅ x 𝜔 suc z = suc x)))
264, 8, 12, 16, 18, 25finds 4266 1 (A 𝜔 → (A = ∅ x 𝜔 A = suc x))
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628   = wceq 1242   wcel 1390  wrex 2301  c0 3218  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:  nnsuc  4281
  Copyright terms: Public domain W3C validator