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Theorem disjors 4406
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 2584 . . 3  |-  F/_ i B
2 nfcsb1v 3411 . . 3  |-  F/_ x [_ i  /  x ]_ B
3 csbeq1a 3404 . . 3  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
41, 2, 3cbvdisj 4401 . 2  |-  (Disj  x  e.  A  B  <-> Disj  i  e.  A  [_ i  /  x ]_ B )
5 csbeq1 3398 . . 3  |-  ( i  =  j  ->  [_ i  /  x ]_ B  = 
[_ j  /  x ]_ B )
65disjor 4405 . 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 252 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 187    \/ wo 369    = wceq 1437   A.wral 2775   [_csb 3395    i^i cin 3435   (/)c0 3761  Disj wdisj 4391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-in 3443  df-nul 3762  df-disj 4392
This theorem is referenced by:  disji2  4407  disjprg  4416  disjxiun  4417  disjxun  4418  iundisj2  22488  disji2f  28176  disjpreima  28183  disjxpin  28187  iundisj2f  28189  disjunsn  28193  iundisj2fi  28366  disjxp1  37273  disjinfi  37317
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