Theorem suc0 6389

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 6318	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4107	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4343	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2758	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1541   ∪ cun 3895  ∅c0 4282  {csn 4575  suc csuc 6314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3900  df-un 3902  df-nul 4283  df-suc 6318

This theorem is referenced by:  df1o2  8398  axdc3lem4  10350