Theorem sucdifsn 39085

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 6370	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	difeq1i 4085	. 2 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
3		sucdifsn2 39084	. 2 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
4	2, 3	eqtri 2793	1 ⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴

Syntax hints:   = wceq 1568   ∖ cdif 3910   ∪ cun 3911  {csn 4594  suc csuc 6366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-reg 9557

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-nul 4295  df-sn 4595  df-suc 6370

This theorem is referenced by:  partsuc  39482