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Theorem disji2 4026
Description: Property of a disjoint collection: if  B ( X )  =  C and  B ( Y )  =  D, and  X  =/=  Y, then  C and  D are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disji.1  |-  ( x  =  X  ->  B  =  C )
disji.2  |-  ( x  =  Y  ->  B  =  D )
Assertion
Ref Expression
disji2  |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A
)  /\  X  =/=  Y )  ->  ( C  i^i  D )  =  (/) )
Distinct variable groups:    x, A    x, C    x, D    x, X    x, Y
Allowed substitution hint:    B( x)

Proof of Theorem disji2
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ne 2461 . . 3  |-  ( X  =/=  Y  <->  -.  X  =  Y )
2 disjors 4025 . . . . . 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 eqeq1 2302 . . . . . . . 8  |-  ( y  =  X  ->  (
y  =  z  <->  X  =  z ) )
4 nfcv 2432 . . . . . . . . . . 11  |-  F/_ x X
5 nfcv 2432 . . . . . . . . . . 11  |-  F/_ x C
6 disji.1 . . . . . . . . . . 11  |-  ( x  =  X  ->  B  =  C )
74, 5, 6csbhypf 3129 . . . . . . . . . 10  |-  ( y  =  X  ->  [_ y  /  x ]_ B  =  C )
87ineq1d 3382 . . . . . . . . 9  |-  ( y  =  X  ->  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  ( C  i^i  [_ z  /  x ]_ B ) )
98eqeq1d 2304 . . . . . . . 8  |-  ( y  =  X  ->  (
( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( C  i^i  [_ z  /  x ]_ B )  =  (/) ) )
103, 9orbi12d 690 . . . . . . 7  |-  ( y  =  X  ->  (
( y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) ) 
<->  ( X  =  z  \/  ( C  i^i  [_ z  /  x ]_ B )  =  (/) ) ) )
11 eqeq2 2305 . . . . . . . 8  |-  ( z  =  Y  ->  ( X  =  z  <->  X  =  Y ) )
12 nfcv 2432 . . . . . . . . . . 11  |-  F/_ x Y
13 nfcv 2432 . . . . . . . . . . 11  |-  F/_ x D
14 disji.2 . . . . . . . . . . 11  |-  ( x  =  Y  ->  B  =  D )
1512, 13, 14csbhypf 3129 . . . . . . . . . 10  |-  ( z  =  Y  ->  [_ z  /  x ]_ B  =  D )
1615ineq2d 3383 . . . . . . . . 9  |-  ( z  =  Y  ->  ( C  i^i  [_ z  /  x ]_ B )  =  ( C  i^i  D ) )
1716eqeq1d 2304 . . . . . . . 8  |-  ( z  =  Y  ->  (
( C  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( C  i^i  D )  =  (/) ) )
1811, 17orbi12d 690 . . . . . . 7  |-  ( z  =  Y  ->  (
( X  =  z  \/  ( C  i^i  [_ z  /  x ]_ B )  =  (/) ) 
<->  ( X  =  Y  \/  ( C  i^i  D )  =  (/) ) ) )
1910, 18rspc2v 2903 . . . . . 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  \/  ( C  i^i  D )  =  (/) ) ) )
202, 19syl5bi 208 . . . . 5  |-  ( ( X  e.  A  /\  Y  e.  A )  ->  (Disj  x  e.  A B  ->  ( X  =  Y  \/  ( C  i^i  D )  =  (/) ) ) )
2120impcom 419 . . . 4  |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A
) )  ->  ( X  =  Y  \/  ( C  i^i  D )  =  (/) ) )
2221ord 366 . . 3  |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A
) )  ->  ( -.  X  =  Y  ->  ( C  i^i  D
)  =  (/) ) )
231, 22syl5bi 208 . 2  |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A
) )  ->  ( X  =/=  Y  ->  ( C  i^i  D )  =  (/) ) )
24233impia 1148 1  |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A
)  /\  X  =/=  Y )  ->  ( C  i^i  D )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   [_csb 3094    i^i cin 3164   (/)c0 3468  Disj wdisj 4009
This theorem is referenced by:  disji  4027  disjxiun  4036  voliunlem1  18923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-in 3172  df-nul 3469  df-disj 4010
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