Theorem sucdifsn2 39194

Description: Absorption of union with a singleton by difference. (Contributed by Peter Mazsa, 24-Jul-2024.)

Assertion

Ref	Expression
sucdifsn2	⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴

Proof of Theorem sucdifsn2

Step	Hyp	Ref	Expression
1		disjcsn 9579	. 2 ⊢ (𝐴 ∩ {𝐴}) = ∅
2		undif5 4447	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ → ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905  ∅c0 4286  {csn 4591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-reg 9561

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-nul 4287  df-sn 4592

This theorem is used by:  sucdifsn  39195  partsuc2  39591