Theorem bj-disjcsn 35068

Description: A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 32616 and does not depend on df-ne 2943. (Contributed by BJ, 4-Apr-2019.)

Assertion

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

Proof of Theorem bj-disjcsn

Step	Hyp	Ref	Expression
1		elirr 9286	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4644	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 230	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108   ∩ cin 3882  ∅c0 4253  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-reg 9281

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-sn 4559  df-pr 4561

This theorem is referenced by: (None)