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

Theorem nn0suc 4447
 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 2101 . . 3 (𝑦 = ∅ → (𝑦 = ∅ ↔ ∅ = ∅))
2 eqeq1 2101 . . . 4 (𝑦 = ∅ → (𝑦 = suc 𝑥 ↔ ∅ = suc 𝑥))
32rexbidv 2392 . . 3 (𝑦 = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω ∅ = suc 𝑥))
41, 3orbi12d 745 . 2 (𝑦 = ∅ → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (∅ = ∅ ∨ ∃𝑥 ∈ ω ∅ = suc 𝑥)))
5 eqeq1 2101 . . 3 (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
6 eqeq1 2101 . . . 4 (𝑦 = 𝑧 → (𝑦 = suc 𝑥𝑧 = suc 𝑥))
76rexbidv 2392 . . 3 (𝑦 = 𝑧 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝑧 = suc 𝑥))
85, 7orbi12d 745 . 2 (𝑦 = 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (𝑧 = ∅ ∨ ∃𝑥 ∈ ω 𝑧 = suc 𝑥)))
9 eqeq1 2101 . . 3 (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
10 eqeq1 2101 . . . 4 (𝑦 = suc 𝑧 → (𝑦 = suc 𝑥 ↔ suc 𝑧 = suc 𝑥))
1110rexbidv 2392 . . 3 (𝑦 = suc 𝑧 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥))
129, 11orbi12d 745 . 2 (𝑦 = suc 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (suc 𝑧 = ∅ ∨ ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)))
13 eqeq1 2101 . . 3 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
14 eqeq1 2101 . . . 4 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
1514rexbidv 2392 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
1613, 15orbi12d 745 . 2 (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑥 ∈ ω 𝑦 = suc 𝑥) ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
17 eqid 2095 . . 3 ∅ = ∅
1817orci 688 . 2 (∅ = ∅ ∨ ∃𝑥 ∈ ω ∅ = suc 𝑥)
19 eqid 2095 . . . . 5 suc 𝑧 = suc 𝑧
20 suceq 4253 . . . . . . 7 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
2120eqeq2d 2106 . . . . . 6 (𝑥 = 𝑧 → (suc 𝑧 = suc 𝑥 ↔ suc 𝑧 = suc 𝑧))
2221rspcev 2736 . . . . 5 ((𝑧 ∈ ω ∧ suc 𝑧 = suc 𝑧) → ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)
2319, 22mpan2 417 . . . 4 (𝑧 ∈ ω → ∃𝑥 ∈ ω suc 𝑧 = suc 𝑥)
2423olcd 691 . . 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 4443 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 667   = wceq 1296   ∈ wcel 1445  ∃wrex 2371  ∅c0 3302  suc csuc 4216  ωcom 4433 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-iinf 4431 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-int 3711  df-suc 4222  df-iom 4434 This theorem is referenced by:  nnsuc  4458  nnpredcl  4464  frecabcl  6202  nnsucuniel  6296  nneneq  6653  phpm  6661  dif1enen  6676  fin0  6681  fin0or  6682  diffisn  6689
 Copyright terms: Public domain W3C validator