Theorem suc0 4514

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 4474	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 3353	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 3530	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2256	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1398   ∪ cun 3199  ∅c0 3496  {csn 3673  suc csuc 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-un 3205  df-nul 3497  df-suc 4474

This theorem is referenced by:  ordtriexmidlem  4623  ordtri2orexmid  4627  2ordpr  4628  onsucsssucexmid  4631  onsucelsucexmid  4634  ordsoexmid  4666  ordtri2or2exmid  4675  ontri2orexmidim  4676  nnregexmid  4725  omsinds  4726  tfr0dm  6531  df1o2  6639  nninfsellemdc  16719