Theorem bj-disjcsn 33061

Description: A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 30931 and does not depend on df-ne 2824. (Contributed by BJ, 4-Apr-2019.)

Assertion

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

Proof of Theorem bj-disjcsn

Step	Hyp	Ref	Expression
1		elirr 8543	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4278	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 221	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   = wceq 1523   ∈ wcel 2030   ∩ cin 3606  ∅c0 3948  {csn 4210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-nul 3949  df-sn 4211  df-pr 4213

This theorem is referenced by: (None)