Theorem suc0 6428

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 6359	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4149	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4386	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2763	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1541   ∪ cun 3942  ∅c0 4318  {csn 4622  suc csuc 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3947  df-un 3949  df-nul 4319  df-suc 6359

This theorem is referenced by:  df1o2  8455  axdc3lem4  10430