Theorem suc0 4456

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 4416	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 3316	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 3493	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2229	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1372   ∪ cun 3163  ∅c0 3459  {csn 3632  suc csuc 4410

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-un 3169  df-nul 3460  df-suc 4416

This theorem is referenced by:  ordtriexmidlem  4565  ordtri2orexmid  4569  2ordpr  4570  onsucsssucexmid  4573  onsucelsucexmid  4576  ordsoexmid  4608  ordtri2or2exmid  4617  ontri2orexmidim  4618  nnregexmid  4667  omsinds  4668  tfr0dm  6398  df1o2  6505  nninfsellemdc  15811