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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
StepHypRef Expression
1 elirr 8113 . 2  |-  -.  A  e.  A
2 disjsn 4032 . 2  |-  ( ( A  i^i  { A } )  =  (/)  <->  -.  A  e.  A )
31, 2mpbir 213 1  |-  ( A  i^i  { A }
)  =  (/)
Colors of variables: wff setvar class
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)
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