Theorem sucdifsn 38262

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 6363	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	difeq1i 4102	. 2 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
3		sucdifsn2 38261	. 2 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
4	2, 3	eqtri 2759	1 ⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴

Syntax hints:   = wceq 1540   ∖ cdif 3928   ∪ cun 3929  {csn 4606  suc csuc 6359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-pr 5407  ax-reg 9611

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-nul 4314  df-sn 4607  df-pr 4609  df-suc 6363

This theorem is referenced by:  partsuc  38803