Theorem suc0 4508

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 4468	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 3351	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 3528	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2256	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1397   ∪ cun 3198  ∅c0 3494  {csn 3669  suc csuc 4462

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-nul 3495  df-suc 4468

This theorem is referenced by:  ordtriexmidlem  4617  ordtri2orexmid  4621  2ordpr  4622  onsucsssucexmid  4625  onsucelsucexmid  4628  ordsoexmid  4660  ordtri2or2exmid  4669  ontri2orexmidim  4670  nnregexmid  4719  omsinds  4720  tfr0dm  6487  df1o2  6595  nninfsellemdc  16612