Theorem bj-disjcsn 34648

Description: A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 33935. (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 8042	. 2 |- -. A e. A
2		disjsn 4092	. 2 |- ( ( A i^i { A } ) = (/) <-> -. A e. A )
3	1, 2	mpbir 209	1 |- ( A i^i { A } ) = (/)

Syntax hints:   -. wn 3   = wceq 1395   e. wcel 1819   i^i cin 3470   (/)c0 3793   {csn 4032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-reg 8036

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-nul 3794  df-sn 4033  df-pr 4035

This theorem is referenced by: (None)