Theorem suc0 6442

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 6370	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4112	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4351	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2792	1 ⊢ suc ∅ = {∅}

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3904  ∅c0 4286  {csn 4591  suc csuc 6366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3909  df-un 3911  df-nul 4287  df-suc 6370

This theorem is used by:  df1o2  8466  axdc3lem4  10452