Theorem sucdifsn2 39560

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 9617	. 2 ⊢ (𝐴 ∩ {𝐴}) = ∅
2		undif5 4443	. 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 1569   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-reg 9599

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-sn 4589

This theorem is used by:  sucdifsn  39561  partsuc2  39957