Theorem suc0 4551

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 4511	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 3373	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 3556	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2263	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1402   ∪ cun 3218  ∅c0 3520  {csn 3705  suc csuc 4505

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-suc 4511

This theorem is referenced by:  ordtriexmidlem  4661  ordtri2orexmid  4665  2ordpr  4666  onsucsssucexmid  4669  onsucelsucexmid  4672  ordsoexmid  4704  ordtri2or2exmid  4713  ontri2orexmidim  4714  nnregexmid  4763  omsinds  4764  tfr0dm  6583  df1o2  6691  nninfsellemdc  16958