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Theorem disjnf 27256
Description: In case  x is not free in  B, disjointness is not so interesting since it reduces to cases where  A is a singleton. (Google Groups discussion with Peter Masza) (Contributed by Thierry Arnoux, 26-Jul-2018.)
Assertion
Ref Expression
disjnf  |-  (Disj  x  e.  A  B  <->  ( B  =  (/)  \/  E* x  x  e.  A )
)
Distinct variable groups:    x, A    x, B

Proof of Theorem disjnf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqidd 2468 . . . . 5  |-  ( x  =  y  ->  B  =  B )
21disjor 4437 . . . 4  |-  (Disj  x  e.  A  B  <->  A. x  e.  A  A. y  e.  A  ( x  =  y  \/  ( B  i^i  B )  =  (/) ) )
3 orcom 387 . . . . . . 7  |-  ( ( x  =  y  \/  ( B  i^i  B
)  =  (/) )  <->  ( ( B  i^i  B )  =  (/)  \/  x  =  y ) )
43ralbii 2898 . . . . . 6  |-  ( A. y  e.  A  (
x  =  y  \/  ( B  i^i  B
)  =  (/) )  <->  A. y  e.  A  ( ( B  i^i  B )  =  (/)  \/  x  =  y ) )
5 r19.32v 3012 . . . . . 6  |-  ( A. y  e.  A  (
( B  i^i  B
)  =  (/)  \/  x  =  y )  <->  ( ( B  i^i  B )  =  (/)  \/  A. y  e.  A  x  =  y ) )
64, 5bitri 249 . . . . 5  |-  ( A. y  e.  A  (
x  =  y  \/  ( B  i^i  B
)  =  (/) )  <->  ( ( B  i^i  B )  =  (/)  \/  A. y  e.  A  x  =  y ) )
76ralbii 2898 . . . 4  |-  ( A. x  e.  A  A. y  e.  A  (
x  =  y  \/  ( B  i^i  B
)  =  (/) )  <->  A. x  e.  A  ( ( B  i^i  B )  =  (/)  \/  A. y  e.  A  x  =  y ) )
8 r19.32v 3012 . . . 4  |-  ( A. x  e.  A  (
( B  i^i  B
)  =  (/)  \/  A. y  e.  A  x  =  y )  <->  ( ( B  i^i  B )  =  (/)  \/  A. x  e.  A  A. y  e.  A  x  =  y ) )
92, 7, 83bitri 271 . . 3  |-  (Disj  x  e.  A  B  <->  ( ( B  i^i  B )  =  (/)  \/  A. x  e.  A  A. y  e.  A  x  =  y ) )
10 inidm 3712 . . . . 5  |-  ( B  i^i  B )  =  B
1110eqeq1i 2474 . . . 4  |-  ( ( B  i^i  B )  =  (/)  <->  B  =  (/) )
1211orbi1i 520 . . 3  |-  ( ( ( B  i^i  B
)  =  (/)  \/  A. x  e.  A  A. y  e.  A  x  =  y )  <->  ( B  =  (/)  \/  A. x  e.  A  A. y  e.  A  x  =  y ) )
139, 12bitri 249 . 2  |-  (Disj  x  e.  A  B  <->  ( B  =  (/)  \/  A. x  e.  A  A. y  e.  A  x  =  y ) )
14 moel 27205 . . 3  |-  ( E* x  x  e.  A  <->  A. x  e.  A  A. y  e.  A  x  =  y )
1514orbi2i 519 . 2  |-  ( ( B  =  (/)  \/  E* x  x  e.  A
)  <->  ( B  =  (/)  \/  A. x  e.  A  A. y  e.  A  x  =  y ) )
1613, 15bitr4i 252 1  |-  (Disj  x  e.  A  B  <->  ( B  =  (/)  \/  E* x  x  e.  A )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    = wceq 1379    e. wcel 1767   E*wmo 2276   A.wral 2817    i^i cin 3480   (/)c0 3790  Disj wdisj 4423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rmo 2825  df-v 3120  df-dif 3484  df-in 3488  df-nul 3791  df-disj 4424
This theorem is referenced by: (None)
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