Theorem suc0 5716

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 5646	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 3719	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 3919	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2636	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1475   ∪ cun 3538  ∅c0 3874  {csn 4125  suc csuc 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-suc 5646

This theorem is referenced by:  df1o2  7459  axdc3lem4  9158