Theorem disjcsn 9570

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 9560	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4676	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 234	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2142   ∩ cin 3903  ∅c0 4285  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-reg 9552

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3456  df-dif 3907  df-in 3911  df-nul 4286  df-sn 4589

This theorem is used by:  bnj927  35167  bnj535  35287  sucdifsn2  39162  ressucdifsn2  39164