Theorem disjcsn 9673

Description: A class is disjoint from its singleton. A consequence of regularity. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 4-Apr-2019.)

Assertion

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

Proof of Theorem disjcsn

Step	Hyp	Ref	Expression
1		elirr 9666	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4736	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 231	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2108   ∩ cin 3975  ∅c0 4352  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-pr 5447  ax-reg 9661

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-nul 4353  df-sn 4649  df-pr 4651

This theorem is referenced by:  bnj927  34745  bnj535  34866  sucdifsn2  38193  ressucdifsn2  38199