Theorem sucdifsn2 38806

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 9524	. 2 ⊢ (𝐴 ∩ {𝐴}) = ∅
2		undif5 4424	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ → ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴

Syntax hints:   = wceq 1542   ∖ cdif 3886   ∪ cun 3887   ∩ cin 3888  ∅c0 4273  {csn 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-reg 9507

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-nul 4274  df-sn 4568

This theorem is referenced by:  sucdifsn  38807  partsuc2  39203