Theorem disjcsn 36449

Description: A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 32771 and does not depend on df-ne 2941. (Temporary: as soon as this Mathbox only PR is accepted, I'll open a PR to place this to the main. PM) (Contributed by BJ, 4-Apr-2019.)

Assertion

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

Proof of Theorem disjcsn

Step	Hyp	Ref	Expression
1		elirr 9408	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4651	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 230	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2104   ∩ cin 3891  ∅c0 4262  {csn 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-reg 9403

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-nul 4263  df-sn 4566  df-pr 4568

This theorem is referenced by:  sucdifsn2  36451  ressucdifsn2  36457