Theorem disjcsn 9560

Description: A class is disjoint from its singleton. A consequence of regularity. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 4-Apr-2019.)

Assertion

Ref	Expression
disjcsn	⊢ (𝐴 ∩ {𝐴}) = ∅

Proof of Theorem disjcsn

Step	Hyp	Ref	Expression
1		elirr 9550	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4672	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 233	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   = wceq 1562   ∈ wcel 2144   ∩ cin 3905  ∅c0 4287  {csn 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-reg 9542

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-v 3458  df-dif 3909  df-in 3913  df-nul 4288  df-sn 4585

This theorem is referenced by:  bnj927  35067  bnj535  35187  sucdifsn2  38989  ressucdifsn2  38991