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 4156	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4343	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2787	1 ⊢ suc ∅ = {∅}

This proof depends on syntax axioms:   = wceq 1569   ∪ cun 4148  ∅c0 4277  {csn 4584  suc csuc 6366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4046  df-un 4150  df-nul 4278  df-suc 6370

This theorem is used by:  df1o2  8488  axdc3lem4  10536