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

Theorem suc0 4333
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 4293 . 2 suc ∅ = (∅ ∪ {∅})
2 uncom 3220 . 2 (∅ ∪ {∅}) = ({∅} ∪ ∅)
3 un0 3396 . 2 ({∅} ∪ ∅) = {∅}
41, 2, 33eqtri 2164 1 suc ∅ = {∅}
Colors of variables: wff set class
Syntax hints:   = wceq 1331  cun 3069  c0 3363  {csn 3527  suc csuc 4287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-nul 3364  df-suc 4293
This theorem is referenced by:  ordtriexmidlem  4435  ordtri2orexmid  4438  2ordpr  4439  onsucsssucexmid  4442  onsucelsucexmid  4445  ordsoexmid  4477  ordtri2or2exmid  4486  nnregexmid  4534  omsinds  4535  tfr0dm  6219  df1o2  6326  nninfsellemdc  13236
  Copyright terms: Public domain W3C validator