Theorem disjcsn 9628

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 9618	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4672	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 234	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147   ∩ cin 4092  ∅c0 4281  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-reg 9610

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-dif 4050  df-in 4094  df-nul 4282  df-sn 4589

This theorem is used by:  bnj927  35467  bnj535  35587  sucdifsn2  39608  ressucdifsn2  39610