Theorem suc0 6287

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 6219	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4067	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4305	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2769	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1543   ∪ cun 3864  ∅c0 4237  {csn 4541  suc csuc 6215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-suc 6219

This theorem is referenced by:  df1o2  8214  axdc3lem4  10067