ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  suc0 Unicode version

Theorem suc0 4195
Description: The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
Assertion
Ref Expression
suc0  |-  suc  (/)  =  { (/)
}

Proof of Theorem suc0
StepHypRef Expression
1 df-suc 4155 . 2  |-  suc  (/)  =  (
(/)  u.  { (/) } )
2 uncom 3127 . 2  |-  ( (/)  u. 
{ (/) } )  =  ( { (/) }  u.  (/) )
3 un0 3295 . 2  |-  ( {
(/) }  u.  (/) )  =  { (/) }
41, 2, 33eqtri 2107 1  |-  suc  (/)  =  { (/)
}
Colors of variables: wff set class
Syntax hints:    = wceq 1285    u. cun 2981   (/)c0 3268   {csn 3417   suc csuc 4149
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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613  df-dif 2985  df-un 2987  df-nul 3269  df-suc 4155
This theorem is referenced by:  ordtriexmidlem  4292  ordtri2orexmid  4295  2ordpr  4296  onsucsssucexmid  4299  onsucelsucexmid  4302  ordsoexmid  4334  ordtri2or2exmid  4343  nnregexmid  4389  tfr0dm  5993  df1o2  6099
  Copyright terms: Public domain W3C validator