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Theorem disjors 4389
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 2616 . . 3  |-  F/_ i B
2 nfcsb1v 3414 . . 3  |-  F/_ x [_ i  /  x ]_ B
3 csbeq1a 3407 . . 3  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
41, 2, 3cbvdisj 4383 . 2  |-  (Disj  x  e.  A  B  <-> Disj  i  e.  A  [_ i  /  x ]_ B )
5 csbeq1 3401 . . 3  |-  ( i  =  j  ->  [_ i  /  x ]_ B  = 
[_ j  /  x ]_ B )
65disjor 4387 . 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 1370   A.wral 2799   [_csb 3398    i^i cin 3438   (/)c0 3748  Disj wdisj 4373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-in 3446  df-nul 3749  df-disj 4374
This theorem is referenced by:  disji2  4390  disjprg  4399  disjxiun  4400  disjxun  4401  iundisj2  21173  disji2f  26099  disjpreima  26106  disjxpin  26108  iundisj2f  26110  disjunsn  26114  iundisj2fi  26253
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