Theorem bj-disjcsn 31542

Description: A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 29545. (Contributed by BJ, 4-Apr-2019.)

Assertion

Ref	Expression
bj-disjcsn	|- ( A i^i { A } ) = (/)

Proof of Theorem bj-disjcsn

Step	Hyp	Ref	Expression
1		elirr 8113	. 2 |- -. A e. A
2		disjsn 4032	. 2 |- ( ( A i^i { A } ) = (/) <-> -. A e. A )
3	1, 2	mpbir 213	1 |- ( A i^i { A } ) = (/)

Syntax hints:   -. wn 3   = wceq 1444   e. wcel 1887   i^i cin 3403   (/)c0 3731   {csn 3968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-reg 8107

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-nul 3732  df-sn 3969  df-pr 3971

This theorem is referenced by: (None)