Theorem suc0 6434

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 6363	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4138	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4374	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2763	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1540   ∪ cun 3929  ∅c0 4313  {csn 4606  suc csuc 6359

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-dif 3934  df-un 3936  df-nul 4314  df-suc 6363

This theorem is referenced by:  df1o2  8492  axdc3lem4  10472