Theorem suc0 6340

Description: The successor of the empty set. (Contributed by NM, 1-Feb-2005.)

Assertion

Ref	Expression
suc0	⊢ suc ∅ = {∅}

Proof of Theorem suc0

Step	Hyp	Ref	Expression
1		df-suc 6272	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4087	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4324	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2770	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1539   ∪ cun 3885  ∅c0 4256  {csn 4561  suc csuc 6268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-suc 6272

This theorem is referenced by:  df1o2  8304  axdc3lem4  10209