Theorem disjcsn 9516

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 9508	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4669	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 231	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114   ∩ cin 3901  ∅c0 4286  {csn 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-pr 5378  ax-reg 9501

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3443  df-dif 3905  df-in 3909  df-nul 4287  df-sn 4582

This theorem is referenced by:  bnj927  34927  bnj535  35048  sucdifsn2  38688  ressucdifsn2  38690