Theorem suc0 6459

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 6390	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4158	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4394	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2769	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1540   ∪ cun 3949  ∅c0 4333  {csn 4626  suc csuc 6386

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-suc 6390

This theorem is referenced by:  df1o2  8513  axdc3lem4  10493  pw2bday  28418