Theorem sucdifsn 37750

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 6380	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	difeq1i 4118	. 2 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
3		sucdifsn2 37749	. 2 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
4	2, 3	eqtri 2756	1 ⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴

Syntax hints:   = wceq 1533   ∖ cdif 3946   ∪ cun 3947  {csn 4632  suc csuc 6376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-pr 5433  ax-reg 9625

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-nul 4327  df-sn 4633  df-pr 4635  df-suc 6380

This theorem is referenced by:  partsuc  38292