Theorem disjcsn 9585

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 9575	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4675	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 234	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145   ∩ cin 3901  ∅c0 4282  {csn 4587

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-reg 9567

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3455  df-dif 3905  df-in 3909  df-nul 4283  df-sn 4588

This theorem is used by:  bnj927  35266  bnj535  35386  sucdifsn2  39220  ressucdifsn2  39222