Theorem suc0 6461

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 6392	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4168	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4400	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2767	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1537   ∪ cun 3961  ∅c0 4339  {csn 4631  suc csuc 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-suc 6392

This theorem is referenced by:  df1o2  8512  axdc3lem4  10491  pw2bday  28433