Theorem suc0 6402

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 6331	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4112	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4348	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2764	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1542   ∪ cun 3901  ∅c0 4287  {csn 4582  suc csuc 6327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-un 3908  df-nul 4288  df-suc 6331

This theorem is referenced by:  df1o2  8414  axdc3lem4  10375