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Theorem disji2 4354
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 2579 . . 3  |-  ( X  =/=  Y  <->  -.  X  =  Y )
2 disjors 4353 . . . . . 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 2386 . . . . . . . 8  |-  ( y  =  X  ->  (
y  =  z  <->  X  =  z ) )
4 nfcv 2544 . . . . . . . . . . 11  |-  F/_ x X
5 nfcv 2544 . . . . . . . . . . 11  |-  F/_ x C
6 disji.1 . . . . . . . . . . 11  |-  ( x  =  X  ->  B  =  C )
74, 5, 6csbhypf 3367 . . . . . . . . . 10  |-  ( y  =  X  ->  [_ y  /  x ]_ B  =  C )
87ineq1d 3613 . . . . . . . . 9  |-  ( y  =  X  ->  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  ( C  i^i  [_ z  /  x ]_ B ) )
98eqeq1d 2384 . . . . . . . 8  |-  ( y  =  X  ->  (
( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( C  i^i  [_ z  /  x ]_ B )  =  (/) ) )
103, 9orbi12d 707 . . . . . . 7  |-  ( y  =  X  ->  (
( y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) ) 
<->  ( X  =  z  \/  ( C  i^i  [_ z  /  x ]_ B )  =  (/) ) ) )
11 eqeq2 2397 . . . . . . . 8  |-  ( z  =  Y  ->  ( X  =  z  <->  X  =  Y ) )
12 nfcv 2544 . . . . . . . . . . 11  |-  F/_ x Y
13 nfcv 2544 . . . . . . . . . . 11  |-  F/_ x D
14 disji.2 . . . . . . . . . . 11  |-  ( x  =  Y  ->  B  =  D )
1512, 13, 14csbhypf 3367 . . . . . . . . . 10  |-  ( z  =  Y  ->  [_ z  /  x ]_ B  =  D )
1615ineq2d 3614 . . . . . . . . 9  |-  ( z  =  Y  ->  ( C  i^i  [_ z  /  x ]_ B )  =  ( C  i^i  D ) )
1716eqeq1d 2384 . . . . . . . 8  |-  ( z  =  Y  ->  (
( C  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( C  i^i  D )  =  (/) ) )
1811, 17orbi12d 707 . . . . . . 7  |-  ( z  =  Y  ->  (
( X  =  z  \/  ( C  i^i  [_ z  /  x ]_ B )  =  (/) ) 
<->  ( X  =  Y  \/  ( C  i^i  D )  =  (/) ) ) )
1910, 18rspc2v 3144 . . . . . 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 217 . . . . 5  |-  ( ( X  e.  A  /\  Y  e.  A )  ->  (Disj  x  e.  A  B  ->  ( X  =  Y  \/  ( C  i^i  D )  =  (/) ) ) )
2120impcom 428 . . . 4  |-  ( (Disj  x  e.  A  B  /\  ( X  e.  A  /\  Y  e.  A
) )  ->  ( X  =  Y  \/  ( C  i^i  D )  =  (/) ) )
2221ord 375 . . 3  |-  ( (Disj  x  e.  A  B  /\  ( X  e.  A  /\  Y  e.  A
) )  ->  ( -.  X  =  Y  ->  ( C  i^i  D
)  =  (/) ) )
231, 22syl5bi 217 . 2  |-  ( (Disj  x  e.  A  B  /\  ( X  e.  A  /\  Y  e.  A
) )  ->  ( X  =/=  Y  ->  ( C  i^i  D )  =  (/) ) )
24233impia 1191 1  |-  ( (Disj  x  e.  A  B  /\  ( X  e.  A  /\  Y  e.  A
)  /\  X  =/=  Y )  ->  ( C  i^i  D )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   [_csb 3348    i^i cin 3388   (/)c0 3711  Disj wdisj 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-in 3396  df-nul 3712  df-disj 4339
This theorem is referenced by:  disji  4355  disjxiun  4364  voliunlem1  22045
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