Theorem suc0 6470

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 6401	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 4181	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 4417	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2772	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1537   ∪ cun 3974  ∅c0 4352  {csn 4648  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-suc 6401

This theorem is referenced by:  df1o2  8529  axdc3lem4  10522  pw2bday  28436