Theorem disjcsn 9519

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 9509	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4646	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 233	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   = wceq 1548   ∈ wcel 2121   ∩ cin 3884  ∅c0 4264  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-reg 9501

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-v 3435  df-dif 3888  df-in 3892  df-nul 4265  df-sn 4559

This theorem is referenced by:  bnj927  34967  bnj535  35087  sucdifsn2  38867  ressucdifsn2  38869