Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sucdifsn Structured version   Visualization version   GIF version

Theorem sucdifsn 36910
Description: The difference between the successor and the singleton of a class is the class. (Contributed by Peter Mazsa, 20-Sep-2024.)
Assertion
Ref Expression
sucdifsn (suc 𝐴 ∖ {𝐴}) = 𝐴

Proof of Theorem sucdifsn
StepHypRef Expression
1 df-suc 6359 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
21difeq1i 4114 . 2 (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
3 sucdifsn2 36909 . 2 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
42, 3eqtri 2759 1 (suc 𝐴 ∖ {𝐴}) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  cdif 3941  cun 3942  {csn 4622  suc csuc 6355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-pr 5420  ax-reg 9569
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-nul 4319  df-sn 4623  df-pr 4625  df-suc 6359
This theorem is referenced by:  partsuc  37455
  Copyright terms: Public domain W3C validator