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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
StepHypRef Expression
1 elirr 8042 . 2  |-  -.  A  e.  A
2 disjsn 4092 . 2  |-  ( ( A  i^i  { A } )  =  (/)  <->  -.  A  e.  A )
31, 2mpbir 209 1  |-  ( A  i^i  { A }
)  =  (/)
Colors of variables: wff setvar class
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)
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