Theorem sucdifsn 39163

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

Step	Hyp	Ref	Expression
1		df-suc 6366	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	difeq1i 4076	. 2 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
3		sucdifsn2 39162	. 2 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
4	2, 3	eqtri 2785	1 ⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴

This proof depends on syntax axioms:   = wceq 1569   ∖ cdif 3901   ∪ cun 3902  {csn 4588  suc csuc 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-reg 9552

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-nul 4286  df-sn 4589  df-suc 6366

This theorem is used by:  partsuc  39560