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Theorem disjnf 28017
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 inidm 3668 . . . 4  |-  ( B  i^i  B )  =  B
21eqeq1i 2427 . . 3  |-  ( ( B  i^i  B )  =  (/)  <->  B  =  (/) )
32orbi1i 522 . 2  |-  ( ( ( 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 ) )
4 eqidd 2421 . . . 4  |-  ( x  =  y  ->  B  =  B )
54disjor 4402 . . 3  |-  (Disj  x  e.  A  B  <->  A. x  e.  A  A. y  e.  A  ( x  =  y  \/  ( B  i^i  B )  =  (/) ) )
6 orcom 388 . . . . . 6  |-  ( ( x  =  y  \/  ( B  i^i  B
)  =  (/) )  <->  ( ( B  i^i  B )  =  (/)  \/  x  =  y ) )
76ralbii 2854 . . . . 5  |-  ( A. y  e.  A  (
x  =  y  \/  ( B  i^i  B
)  =  (/) )  <->  A. y  e.  A  ( ( B  i^i  B )  =  (/)  \/  x  =  y ) )
8 r19.32v 2972 . . . . 5  |-  ( A. y  e.  A  (
( B  i^i  B
)  =  (/)  \/  x  =  y )  <->  ( ( B  i^i  B )  =  (/)  \/  A. y  e.  A  x  =  y ) )
97, 8bitri 252 . . . 4  |-  ( A. y  e.  A  (
x  =  y  \/  ( B  i^i  B
)  =  (/) )  <->  ( ( B  i^i  B )  =  (/)  \/  A. y  e.  A  x  =  y ) )
109ralbii 2854 . . 3  |-  ( 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 ) )
11 r19.32v 2972 . . 3  |-  ( 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 ) )
125, 10, 113bitri 274 . 2  |-  (Disj  x  e.  A  B  <->  ( ( B  i^i  B )  =  (/)  \/  A. x  e.  A  A. y  e.  A  x  =  y ) )
13 moel 27951 . . 3  |-  ( E* x  x  e.  A  <->  A. x  e.  A  A. y  e.  A  x  =  y )
1413orbi2i 521 . 2  |-  ( ( B  =  (/)  \/  E* x  x  e.  A
)  <->  ( B  =  (/)  \/  A. x  e.  A  A. y  e.  A  x  =  y ) )
153, 12, 143bitr4i 280 1  |-  (Disj  x  e.  A  B  <->  ( B  =  (/)  \/  E* x  x  e.  A )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    \/ wo 369    = wceq 1437    e. wcel 1867   E*wmo 2264   A.wral 2773    i^i cin 3432   (/)c0 3758  Disj wdisj 4388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rmo 2781  df-v 3080  df-dif 3436  df-in 3440  df-nul 3759  df-disj 4389
This theorem is referenced by: (None)
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