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Theorem disji2 3890
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 disjnims 3889 . . 3  |-  (Disj  x  e.  A  B  ->  A. y  e.  A  A. z  e.  A  (
y  =/=  z  -> 
( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) ) )
2 neeq1 2296 . . . . 5  |-  ( y  =  X  ->  (
y  =/=  z  <->  X  =/=  z ) )
3 nfcv 2256 . . . . . . . 8  |-  F/_ x X
4 nfcv 2256 . . . . . . . 8  |-  F/_ x C
5 disji.1 . . . . . . . 8  |-  ( x  =  X  ->  B  =  C )
63, 4, 5csbhypf 3006 . . . . . . 7  |-  ( y  =  X  ->  [_ y  /  x ]_ B  =  C )
76ineq1d 3244 . . . . . 6  |-  ( y  =  X  ->  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  ( C  i^i  [_ z  /  x ]_ B ) )
87eqeq1d 2124 . . . . 5  |-  ( y  =  X  ->  (
( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( C  i^i  [_ z  /  x ]_ B )  =  (/) ) )
92, 8imbi12d 233 . . . 4  |-  ( y  =  X  ->  (
( y  =/=  z  ->  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) )  <->  ( X  =/=  z  ->  ( C  i^i  [_ z  /  x ]_ B )  =  (/) ) ) )
10 neeq2 2297 . . . . 5  |-  ( z  =  Y  ->  ( X  =/=  z  <->  X  =/=  Y ) )
11 nfcv 2256 . . . . . . . 8  |-  F/_ x Y
12 nfcv 2256 . . . . . . . 8  |-  F/_ x D
13 disji.2 . . . . . . . 8  |-  ( x  =  Y  ->  B  =  D )
1411, 12, 13csbhypf 3006 . . . . . . 7  |-  ( z  =  Y  ->  [_ z  /  x ]_ B  =  D )
1514ineq2d 3245 . . . . . 6  |-  ( z  =  Y  ->  ( C  i^i  [_ z  /  x ]_ B )  =  ( C  i^i  D ) )
1615eqeq1d 2124 . . . . 5  |-  ( z  =  Y  ->  (
( C  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( C  i^i  D )  =  (/) ) )
1710, 16imbi12d 233 . . . 4  |-  ( z  =  Y  ->  (
( X  =/=  z  ->  ( C  i^i  [_ z  /  x ]_ B )  =  (/) )  <->  ( X  =/=  Y  ->  ( C  i^i  D )  =  (/) ) ) )
189, 17rspc2v 2774 . . 3  |-  ( ( 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 )  =  (/) ) ) )
191, 18mpan9 277 . 2  |-  ( (Disj  x  e.  A  B  /\  ( X  e.  A  /\  Y  e.  A
) )  ->  ( X  =/=  Y  ->  ( C  i^i  D )  =  (/) ) )
20193impia 1161 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:    -> wi 4    /\ wa 103    /\ w3a 945    = wceq 1314    e. wcel 1463    =/= wne 2283   A.wral 2391   [_csb 2973    i^i cin 3038   (/)c0 3331  Disj wdisj 3874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-in 3045  df-nul 3332  df-disj 3875
This theorem is referenced by: (None)
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