Theorem bj-disjcsn 33795

Description: A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 31687 and does not depend on df-ne 2961. (Contributed by BJ, 4-Apr-2019.)

Assertion

Ref	Expression
bj-disjcsn	⊢ (𝐴 ∩ {𝐴}) = ∅

Proof of Theorem bj-disjcsn

Step	Hyp	Ref	Expression
1		elirr 8854	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4517	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 223	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   = wceq 1508   ∈ wcel 2051   ∩ cin 3821  ∅c0 4172  {csn 4435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182  ax-reg 8849

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-nul 4173  df-sn 4436  df-pr 4438

This theorem is referenced by: (None)