Theorem sucdifsn 38990

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 6354	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	difeq1i 4078	. 2 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
3		sucdifsn2 38989	. 2 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
4	2, 3	eqtri 2787	1 ⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴

Syntax hints:   = wceq 1562   ∖ cdif 3903   ∪ cun 3904  {csn 4584  suc csuc 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-reg 9542

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-nul 4288  df-sn 4585  df-suc 6354

This theorem is referenced by:  partsuc  39387