Theorem bj-disjcsn 34704

Description: A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 32249 and does not depend on df-ne 2953. (Contributed by BJ, 4-Apr-2019.)

Assertion

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

Proof of Theorem bj-disjcsn

Step	Hyp	Ref	Expression
1		elirr 9108	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4608	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 234	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2112   ∩ cin 3860  ∅c0 4228  {csn 4526

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303  ax-reg 9103

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3864  df-un 3866  df-in 3868  df-nul 4229  df-sn 4527  df-pr 4529

This theorem is referenced by: (None)