Theorem suc0 6325

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 6257	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4083	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4321	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2770	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1539   ∪ cun 3881  ∅c0 4253  {csn 4558  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-suc 6257

This theorem is referenced by:  df1o2  8279  axdc3lem4  10140