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Theorem sucdifsn 37100
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 6370 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
21difeq1i 4118 . 2 (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
3 sucdifsn2 37099 . 2 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
42, 3eqtri 2760 1 (suc 𝐴 ∖ {𝐴}) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  cdif 3945  cun 3946  {csn 4628  suc csuc 6366
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 2703  ax-sep 5299  ax-pr 5427  ax-reg 9586
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 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-nul 4323  df-sn 4629  df-pr 4631  df-suc 6370
This theorem is referenced by:  partsuc  37645
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