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Theorem disji2 4276
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 2606 . . 3  |-  ( X  =/=  Y  <->  -.  X  =  Y )
2 disjors 4275 . . . . . 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 2447 . . . . . . . 8  |-  ( y  =  X  ->  (
y  =  z  <->  X  =  z ) )
4 nfcv 2577 . . . . . . . . . . 11  |-  F/_ x X
5 nfcv 2577 . . . . . . . . . . 11  |-  F/_ x C
6 disji.1 . . . . . . . . . . 11  |-  ( x  =  X  ->  B  =  C )
74, 5, 6csbhypf 3304 . . . . . . . . . 10  |-  ( y  =  X  ->  [_ y  /  x ]_ B  =  C )
87ineq1d 3548 . . . . . . . . 9  |-  ( y  =  X  ->  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  ( C  i^i  [_ z  /  x ]_ B ) )
98eqeq1d 2449 . . . . . . . 8  |-  ( y  =  X  ->  (
( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( C  i^i  [_ z  /  x ]_ B )  =  (/) ) )
103, 9orbi12d 704 . . . . . . 7  |-  ( y  =  X  ->  (
( y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) ) 
<->  ( X  =  z  \/  ( C  i^i  [_ z  /  x ]_ B )  =  (/) ) ) )
11 eqeq2 2450 . . . . . . . 8  |-  ( z  =  Y  ->  ( X  =  z  <->  X  =  Y ) )
12 nfcv 2577 . . . . . . . . . . 11  |-  F/_ x Y
13 nfcv 2577 . . . . . . . . . . 11  |-  F/_ x D
14 disji.2 . . . . . . . . . . 11  |-  ( x  =  Y  ->  B  =  D )
1512, 13, 14csbhypf 3304 . . . . . . . . . 10  |-  ( z  =  Y  ->  [_ z  /  x ]_ B  =  D )
1615ineq2d 3549 . . . . . . . . 9  |-  ( z  =  Y  ->  ( C  i^i  [_ z  /  x ]_ B )  =  ( C  i^i  D ) )
1716eqeq1d 2449 . . . . . . . 8  |-  ( z  =  Y  ->  (
( C  i^i  [_ z  /  x ]_ B )  =  (/)  <->  ( C  i^i  D )  =  (/) ) )
1811, 17orbi12d 704 . . . . . . 7  |-  ( z  =  Y  ->  (
( X  =  z  \/  ( C  i^i  [_ z  /  x ]_ B )  =  (/) ) 
<->  ( X  =  Y  \/  ( C  i^i  D )  =  (/) ) ) )
1910, 18rspc2v 3076 . . . . . 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 430 . . . 4  |-  ( (Disj  x  e.  A  B  /\  ( X  e.  A  /\  Y  e.  A
) )  ->  ( X  =  Y  \/  ( C  i^i  D )  =  (/) ) )
2221ord 377 . . 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 1179 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 368    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   [_csb 3285    i^i cin 3324   (/)c0 3634  Disj wdisj 4259
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 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-in 3332  df-nul 3635  df-disj 4260
This theorem is referenced by:  disji  4277  disjxiun  4286  voliunlem1  20931
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