Theorem suc0 6436

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 6364	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4120	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4357	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2796	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1567   ∪ cun 3911  ∅c0 4294  {csn 4591  suc csuc 6360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-un 3918  df-nul 4295  df-suc 6364

This theorem is referenced by:  df1o2  8456  axdc3lem4  10433