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Theorem disjors 4439
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 2629 . . 3  |-  F/_ i B
2 nfcsb1v 3456 . . 3  |-  F/_ x [_ i  /  x ]_ B
3 csbeq1a 3449 . . 3  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
41, 2, 3cbvdisj 4433 . 2  |-  (Disj  x  e.  A  B  <-> Disj  i  e.  A  [_ i  /  x ]_ B )
5 csbeq1 3443 . . 3  |-  ( i  =  j  ->  [_ i  /  x ]_ B  = 
[_ j  /  x ]_ B )
65disjor 4437 . 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 1379   A.wral 2817   [_csb 3440    i^i cin 3480   (/)c0 3790  Disj wdisj 4423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-in 3488  df-nul 3791  df-disj 4424
This theorem is referenced by:  disji2  4440  disjprg  4449  disjxiun  4450  disjxun  4451  iundisj2  21827  disji2f  27261  disjpreima  27268  disjxpin  27270  iundisj2f  27272  disjunsn  27276  iundisj2fi  27425
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