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Theorem disji2f 27097
Description: Property of a disjoint collection: if  B ( x )  =  C and  B ( Y )  =  D, and  x  =/=  Y, then  B and  C are disjoint. (Contributed by Thierry Arnoux, 30-Dec-2016.)
Hypotheses
Ref Expression
disjif.1  |-  F/_ x C
disjif.2  |-  ( x  =  Y  ->  B  =  C )
Assertion
Ref Expression
disji2f  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
)  /\  x  =/=  Y )  ->  ( B  i^i  C )  =  (/) )
Distinct variable groups:    x, A    x, Y
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem disji2f
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ne 2657 . . 3  |-  ( x  =/=  Y  <->  -.  x  =  Y )
2 disjors 4426 . . . . . 6  |-  (Disj  x  e.  A  B  <->  A. y  e.  A  A. z  e.  A  ( y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) ) )
3 equequ1 1742 . . . . . . . 8  |-  ( y  =  x  ->  (
y  =  z  <->  x  =  z ) )
4 csbeq1 3431 . . . . . . . . . . 11  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
5 csbid 3436 . . . . . . . . . . 11  |-  [_ x  /  x ]_ B  =  B
64, 5syl6eq 2517 . . . . . . . . . 10  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
76ineq1d 3692 . . . . . . . . 9  |-  ( y  =  x  ->  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  ( B  i^i  [_ z  /  x ]_ B ) )
87eqeq1d 2462 . . . . . . . 8  |-  ( y  =  x  ->  (
( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( B  i^i  [_ z  /  x ]_ B )  =  (/) ) )
93, 8orbi12d 709 . . . . . . 7  |-  ( y  =  x  ->  (
( y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) ) 
<->  ( x  =  z  \/  ( B  i^i  [_ z  /  x ]_ B )  =  (/) ) ) )
10 eqeq2 2475 . . . . . . . 8  |-  ( z  =  Y  ->  (
x  =  z  <->  x  =  Y ) )
11 nfcv 2622 . . . . . . . . . . 11  |-  F/_ x Y
12 disjif.1 . . . . . . . . . . 11  |-  F/_ x C
13 disjif.2 . . . . . . . . . . 11  |-  ( x  =  Y  ->  B  =  C )
1411, 12, 13csbhypf 3447 . . . . . . . . . 10  |-  ( z  =  Y  ->  [_ z  /  x ]_ B  =  C )
1514ineq2d 3693 . . . . . . . . 9  |-  ( z  =  Y  ->  ( B  i^i  [_ z  /  x ]_ B )  =  ( B  i^i  C ) )
1615eqeq1d 2462 . . . . . . . 8  |-  ( z  =  Y  ->  (
( B  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( B  i^i  C )  =  (/) ) )
1710, 16orbi12d 709 . . . . . . 7  |-  ( z  =  Y  ->  (
( x  =  z  \/  ( B  i^i  [_ z  /  x ]_ B )  =  (/) ) 
<->  ( x  =  Y  \/  ( B  i^i  C )  =  (/) ) ) )
189, 17rspc2v 3216 . . . . . 6  |-  ( ( x  e.  A  /\  Y  e.  A )  ->  ( A. y  e.  A  A. z  e.  A  ( y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) )  ->  ( x  =  Y  \/  ( B  i^i  C )  =  (/) ) ) )
192, 18syl5bi 217 . . . . 5  |-  ( ( x  e.  A  /\  Y  e.  A )  ->  (Disj  x  e.  A  B  ->  ( x  =  Y  \/  ( B  i^i  C )  =  (/) ) ) )
2019impcom 430 . . . 4  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
) )  ->  (
x  =  Y  \/  ( B  i^i  C )  =  (/) ) )
2120ord 377 . . 3  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
) )  ->  ( -.  x  =  Y  ->  ( B  i^i  C
)  =  (/) ) )
221, 21syl5bi 217 . 2  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
) )  ->  (
x  =/=  Y  -> 
( B  i^i  C
)  =  (/) ) )
23223impia 1188 1  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
)  /\  x  =/=  Y )  ->  ( B  i^i  C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   F/_wnfc 2608    =/= wne 2655   A.wral 2807   [_csb 3428    i^i cin 3468   (/)c0 3778  Disj wdisj 4410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-in 3476  df-nul 3779  df-disj 4411
This theorem is referenced by:  disjif  27098
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