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

Theorem nn0suc 4432
 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
Distinct variable group:   ,

Proof of Theorem nn0suc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2095 . . 3
2 eqeq1 2095 . . . 4
32rexbidv 2382 . . 3
41, 3orbi12d 743 . 2
5 eqeq1 2095 . . 3
6 eqeq1 2095 . . . 4
76rexbidv 2382 . . 3
85, 7orbi12d 743 . 2
9 eqeq1 2095 . . 3
10 eqeq1 2095 . . . 4
1110rexbidv 2382 . . 3
129, 11orbi12d 743 . 2
13 eqeq1 2095 . . 3
14 eqeq1 2095 . . . 4
1514rexbidv 2382 . . 3
1613, 15orbi12d 743 . 2
17 eqid 2089 . . 3
1817orci 686 . 2
19 eqid 2089 . . . . 5
20 suceq 4238 . . . . . . 7
2120eqeq2d 2100 . . . . . 6
2221rspcev 2723 . . . . 5
2319, 22mpan2 417 . . . 4
2423olcd 689 . . 3
2524a1d 22 . 2
264, 8, 12, 16, 18, 25finds 4428 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 665   wceq 1290   wcel 1439  wrex 2361  c0 3287   csuc 4201  com 4418 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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-iinf 4416 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-uni 3660  df-int 3695  df-suc 4207  df-iom 4419 This theorem is referenced by:  nnsuc  4443  nnpredcl  4449  frecabcl  6178  nnsucuniel  6270  nneneq  6627  phpm  6635  dif1enen  6650  fin0  6655  fin0or  6656  diffisn  6663
 Copyright terms: Public domain W3C validator