Theorem suc0 6440

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 6371	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4154	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4391	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2762	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1539   ∪ cun 3947  ∅c0 4323  {csn 4629  suc csuc 6367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3952  df-un 3954  df-nul 4324  df-suc 6371

This theorem is referenced by:  df1o2  8477  axdc3lem4  10452