Theorem suc0 6268

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 6200	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4132	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4347	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2851	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1536   ∪ cun 3937  ∅c0 4294  {csn 4570  suc csuc 6196

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-dif 3942  df-un 3944  df-nul 4295  df-suc 6200

This theorem is referenced by:  df1o2  8119  axdc3lem4  9878