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Theorem disjors 4442
Description: Two ways to say that a collection  B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjors  |-  (Disj  x  e.  A  B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
Distinct variable groups:    i, j, x, A    B, i, j
Allowed substitution hint:    B( x)

Proof of Theorem disjors
StepHypRef Expression
1 nfcv 2619 . . 3  |-  F/_ i B
2 nfcsb1v 3446 . . 3  |-  F/_ x [_ i  /  x ]_ B
3 csbeq1a 3439 . . 3  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
41, 2, 3cbvdisj 4437 . 2  |-  (Disj  x  e.  A  B  <-> Disj  i  e.  A  [_ i  /  x ]_ B )
5 csbeq1 3433 . . 3  |-  ( i  =  j  ->  [_ i  /  x ]_ B  = 
[_ j  /  x ]_ B )
65disjor 4441 . 2  |-  (Disj  i  e.  A  [_ i  /  x ]_ B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
74, 6bitri 249 1  |-  (Disj  x  e.  A  B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    = wceq 1395   A.wral 2807   [_csb 3430    i^i cin 3470   (/)c0 3793  Disj wdisj 4427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-in 3478  df-nul 3794  df-disj 4428
This theorem is referenced by:  disji2  4443  disjprg  4452  disjxiun  4453  disjxun  4454  iundisj2  22085  disji2f  27577  disjpreima  27584  disjxpin  27587  iundisj2f  27589  disjunsn  27593  iundisj2fi  27762
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