Theorem suc0 4501

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 4461	. 2 ⊢ suc ∅ = (∅ ∪ {∅})
2		uncom 3348	. 2 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
3		un0 3525	. 2 ⊢ ({∅} ∪ ∅) = {∅}
4	1, 2, 3	3eqtri 2254	1 ⊢ suc ∅ = {∅}

Syntax hints:   = wceq 1395   ∪ cun 3195  ∅c0 3491  {csn 3666  suc csuc 4455

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492  df-suc 4461

This theorem is referenced by:  ordtriexmidlem  4610  ordtri2orexmid  4614  2ordpr  4615  onsucsssucexmid  4618  onsucelsucexmid  4621  ordsoexmid  4653  ordtri2or2exmid  4662  ontri2orexmidim  4663  nnregexmid  4712  omsinds  4713  tfr0dm  6466  df1o2  6573  nninfsellemdc  16335