Theorem suc0 6140

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 6072	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4050	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4264	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2823	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1522   ∪ cun 3857  ∅c0 4211  {csn 4472  suc csuc 6068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-dif 3862  df-un 3864  df-nul 4212  df-suc 6072

This theorem is referenced by:  df1o2  7967  axdc3lem4  9721