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Theorem sucdifsn 38241
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 6389 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
21difeq1i 4121 . 2 (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
3 sucdifsn2 38240 . 2 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
42, 3eqtri 2764 1 (suc 𝐴 ∖ {𝐴}) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cdif 3947  cun 3948  {csn 4625  suc csuc 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-pr 5431  ax-reg 9633
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-nul 4333  df-sn 4626  df-pr 4628  df-suc 6389
This theorem is referenced by:  partsuc  38782
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