MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suc0 Structured version   Unicode version

Theorem suc0 4655
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 4587 . 2  |-  suc  (/)  =  (
(/)  u.  { (/) } )
2 uncom 3491 . 2  |-  ( (/)  u. 
{ (/) } )  =  ( { (/) }  u.  (/) )
3 un0 3652 . 2  |-  ( {
(/) }  u.  (/) )  =  { (/) }
41, 2, 33eqtri 2460 1  |-  suc  (/)  =  { (/)
}
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3318   (/)c0 3628   {csn 3814   suc csuc 4583
This theorem is referenced by:  df1o2  6736  axdc3lem4  8333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323  df-un 3325  df-nul 3629  df-suc 4587
  Copyright terms: Public domain W3C validator