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

Theorem suc0 5513
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 5445 . 2  |-  suc  (/)  =  (
(/)  u.  { (/) } )
2 uncom 3610 . 2  |-  ( (/)  u. 
{ (/) } )  =  ( { (/) }  u.  (/) )
3 un0 3787 . 2  |-  ( {
(/) }  u.  (/) )  =  { (/) }
41, 2, 33eqtri 2455 1  |-  suc  (/)  =  { (/)
}
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    u. cun 3434   (/)c0 3761   {csn 3996   suc csuc 5441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-dif 3439  df-un 3441  df-nul 3762  df-suc 5445
This theorem is referenced by:  df1o2  7199  axdc3lem4  8884
  Copyright terms: Public domain W3C validator