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 37618
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 6364 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
21difeq1i 4113 . 2 (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
3 sucdifsn2 37617 . 2 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
42, 3eqtri 2754 1 (suc 𝐴 ∖ {𝐴}) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cdif 3940  cun 3941  {csn 4623  suc csuc 6360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-pr 5420  ax-reg 9589
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-nul 4318  df-sn 4624  df-pr 4626  df-suc 6364
This theorem is referenced by:  partsuc  38163
  Copyright terms: Public domain W3C validator