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

Theorem nn0suc 4381
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 2089 . . 3 (𝑦 = ∅ → (𝑦 = ∅ ↔ ∅ = ∅))
2 eqeq1 2089 . . . 4 (𝑦 = ∅ → (𝑦 = suc 𝑥 ↔ ∅ = suc 𝑥))
32rexbidv 2375 . . 3 (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω ∅ = suc 𝑥))
41, 3orbi12d 740 . 2 (𝑦 = ∅ → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (∅ = ∅ ∨ ∃𝑥 ∈ ω ∅ = suc 𝑥)))
5 eqeq1 2089 . . 3 (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
6 eqeq1 2089 . . . 4 (𝑦 = 𝑧 → (𝑦 = suc 𝑥𝑧 = suc 𝑥))
76rexbidv 2375 . . 3 (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝑧 = suc 𝑥))
85, 7orbi12d 740 . 2 (𝑦 = 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (𝑧 = ∅ ∨ ∃𝑥 ∈ ω 𝑧 = suc 𝑥)))
9 eqeq1 2089 . . 3 (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
10 eqeq1 2089 . . . 4 (𝑦 = suc 𝑧 → (𝑦 = suc 𝑥 ↔ suc 𝑧 = suc 𝑥))
1110rexbidv 2375 . . 3 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥))
129, 11orbi12d 740 . 2 (𝑦 = suc 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (suc 𝑧 = ∅ ∨ ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)))
13 eqeq1 2089 . . 3 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
14 eqeq1 2089 . . . 4 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
1514rexbidv 2375 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
1613, 15orbi12d 740 . 2 (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
17 eqid 2083 . . 3 ∅ = ∅
1817orci 683 . 2 (∅ = ∅ ∨ ∃𝑥 ∈ ω ∅ = suc 𝑥)
19 eqid 2083 . . . . 5 suc 𝑧 = suc 𝑧
20 suceq 4192 . . . . . . 7 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
2120eqeq2d 2094 . . . . . 6 (𝑥 = 𝑧 → (suc 𝑧 = suc 𝑥 ↔ suc 𝑧 = suc 𝑧))
2221rspcev 2712 . . . . 5 ((𝑧 ∈ ω ∧ suc 𝑧 = suc 𝑧) → ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)
2319, 22mpan2 416 . . . 4 (𝑧 ∈ ω → ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)
2423olcd 686 . . 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 4377 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662   = wceq 1285  wcel 1434  wrex 2354  c0 3269  suc csuc 4155  ωcom 4367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-int 3663  df-suc 4161  df-iom 4368
This theorem is referenced by:  nnsuc  4392  frecabcl  6094  nnsucuniel  6186  nneneq  6501  phpm  6509  dif1enen  6524  fin0  6529  fin0or  6530  diffisn  6537
  Copyright terms: Public domain W3C validator