Theorem suc0 5799

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 5729	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 3757	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 3967	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2648	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1483   ∪ cun 3572  ∅c0 3915  {csn 4177  suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-suc 5729

This theorem is referenced by:  df1o2  7572  axdc3lem4  9275