Theorem suc0 6418

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 6347	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4109	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4345	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2788	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1559   ∪ cun 3900  ∅c0 4283  {csn 4579  suc csuc 6343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3905  df-un 3907  df-nul 4284  df-suc 6347

This theorem is referenced by:  df1o2  8438  axdc3lem4  10404