Theorem bj-disjcsn 32936

Description: A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 30805 and does not depend on df-ne 2795. (Contributed by BJ, 4-Apr-2019.)

Assertion

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

Proof of Theorem bj-disjcsn

Step	Hyp	Ref	Expression
1		elirr 8505	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn 4246	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴)
3	1, 2	mpbir 221	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990   ∩ cin 3573  ∅c0 3915  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)