Theorem suc0 6438

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 6366	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4112	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4351	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2790	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1570   ∪ cun 3903  ∅c0 4286  {csn 4589  suc csuc 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3908  df-un 3910  df-nul 4287  df-suc 6366

This theorem is referenced by:  df1o2  8456  axdc3lem4  10432