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Theorem disjif2 28191
Description: Property of a disjoint collection: if  B ( x ) and  B ( Y )  =  D have a common element  Z, then  x  =  Y. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Hypotheses
Ref Expression
disjif2.1  |-  F/_ x A
disjif2.2  |-  F/_ x C
disjif2.3  |-  ( x  =  Y  ->  B  =  C )
Assertion
Ref Expression
disjif2  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
)  /\  ( Z  e.  B  /\  Z  e.  C ) )  ->  x  =  Y )
Distinct variable group:    x, Y
Allowed substitution hints:    A( x)    B( x)    C( x)    Z( x)

Proof of Theorem disjif2
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inelcm 3819 . 2  |-  ( ( Z  e.  B  /\  Z  e.  C )  ->  ( B  i^i  C
)  =/=  (/) )
2 disjif2.1 . . . . . . . 8  |-  F/_ x A
32disjorsf 28190 . . . . . . 7  |-  (Disj  x  e.  A  B  <->  A. y  e.  A  A. z  e.  A  ( y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) ) )
4 equequ1 1867 . . . . . . . . 9  |-  ( y  =  x  ->  (
y  =  z  <->  x  =  z ) )
5 csbeq1 3366 . . . . . . . . . . . 12  |-  ( y  =  x  ->  [_ y  /  x ]_ B  = 
[_ x  /  x ]_ B )
6 csbid 3371 . . . . . . . . . . . 12  |-  [_ x  /  x ]_ B  =  B
75, 6syl6eq 2501 . . . . . . . . . . 11  |-  ( y  =  x  ->  [_ y  /  x ]_ B  =  B )
87ineq1d 3633 . . . . . . . . . 10  |-  ( y  =  x  ->  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  ( B  i^i  [_ z  /  x ]_ B ) )
98eqeq1d 2453 . . . . . . . . 9  |-  ( y  =  x  ->  (
( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( B  i^i  [_ z  /  x ]_ B )  =  (/) ) )
104, 9orbi12d 716 . . . . . . . 8  |-  ( y  =  x  ->  (
( y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) ) 
<->  ( x  =  z  \/  ( B  i^i  [_ z  /  x ]_ B )  =  (/) ) ) )
11 eqeq2 2462 . . . . . . . . 9  |-  ( z  =  Y  ->  (
x  =  z  <->  x  =  Y ) )
12 nfcv 2592 . . . . . . . . . . . 12  |-  F/_ x Y
13 disjif2.2 . . . . . . . . . . . 12  |-  F/_ x C
14 disjif2.3 . . . . . . . . . . . 12  |-  ( x  =  Y  ->  B  =  C )
1512, 13, 14csbhypf 3382 . . . . . . . . . . 11  |-  ( z  =  Y  ->  [_ z  /  x ]_ B  =  C )
1615ineq2d 3634 . . . . . . . . . 10  |-  ( z  =  Y  ->  ( B  i^i  [_ z  /  x ]_ B )  =  ( B  i^i  C ) )
1716eqeq1d 2453 . . . . . . . . 9  |-  ( z  =  Y  ->  (
( B  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( B  i^i  C )  =  (/) ) )
1811, 17orbi12d 716 . . . . . . . 8  |-  ( z  =  Y  ->  (
( x  =  z  \/  ( B  i^i  [_ z  /  x ]_ B )  =  (/) ) 
<->  ( x  =  Y  \/  ( B  i^i  C )  =  (/) ) ) )
1910, 18rspc2v 3159 . . . . . . 7  |-  ( ( 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 )  =  (/) ) ) )
203, 19syl5bi 221 . . . . . 6  |-  ( ( x  e.  A  /\  Y  e.  A )  ->  (Disj  x  e.  A  B  ->  ( x  =  Y  \/  ( B  i^i  C )  =  (/) ) ) )
2120impcom 432 . . . . 5  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
) )  ->  (
x  =  Y  \/  ( B  i^i  C )  =  (/) ) )
2221ord 379 . . . 4  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
) )  ->  ( -.  x  =  Y  ->  ( B  i^i  C
)  =  (/) ) )
2322necon1ad 2641 . . 3  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
) )  ->  (
( B  i^i  C
)  =/=  (/)  ->  x  =  Y ) )
24233impia 1205 . 2  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
)  /\  ( B  i^i  C )  =/=  (/) )  ->  x  =  Y )
251, 24syl3an3 1303 1  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
)  /\  ( Z  e.  B  /\  Z  e.  C ) )  ->  x  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   F/_wnfc 2579    =/= wne 2622   A.wral 2737   [_csb 3363    i^i cin 3403   (/)c0 3731  Disj wdisj 4373
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-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rmo 2745  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-in 3411  df-nul 3732  df-disj 4374
This theorem is referenced by:  disjabrexf  28193
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