Theorem bj-disjcsn 35141

Description: A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 32716 and does not depend on df-ne 2944. (Contributed by BJ, 4-Apr-2019.)

Assertion

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

Proof of Theorem bj-disjcsn

Step	Hyp	Ref	Expression
1		elirr 9356	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4647	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 230	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2106   ∩ cin 3886  ∅c0 4256  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-reg 9351

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-sn 4562  df-pr 4564

This theorem is referenced by: (None)