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Theorem suc0 6140
 Description: The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
Assertion
Ref Expression
suc0 suc ∅ = {∅}

Proof of Theorem suc0
StepHypRef Expression
1 df-suc 6072 . 2 suc ∅ = (∅ ∪ {∅})
2 uncom 4050 . 2 (∅ ∪ {∅}) = ({∅} ∪ ∅)
3 un0 4264 . 2 ({∅} ∪ ∅) = {∅}
41, 2, 33eqtri 2823 1 suc ∅ = {∅}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1522   ∪ cun 3857  ∅c0 4211  {csn 4472  suc csuc 6068 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-dif 3862  df-un 3864  df-nul 4212  df-suc 6072 This theorem is referenced by:  df1o2  7967  axdc3lem4  9721
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