Theorem sucdifsn2 38226

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 9557	. 2 ⊢ (𝐴 ∩ {𝐴}) = ∅
2		undif5 4448	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ → ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴

Syntax hints:   = wceq 1540   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913  ∅c0 4296  {csn 4589

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 2701  ax-sep 5251  ax-pr 5387  ax-reg 9545

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 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-nul 4297  df-sn 4590  df-pr 4592

This theorem is referenced by:  sucdifsn  38227  partsuc2  38771