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Theorem disjors 4200
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 2574 . . 3  |-  F/_ i B
2 nfcsb1v 3285 . . 3  |-  F/_ x [_ i  /  x ]_ B
3 csbeq1a 3261 . . 3  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
41, 2, 3cbvdisj 4194 . 2  |-  (Disj  x  e.  A B  <-> Disj  i  e.  A [_ i  /  x ]_ B )
5 csbeq1 3256 . . 3  |-  ( i  =  j  ->  [_ i  /  x ]_ B  = 
[_ j  /  x ]_ B )
65disjor 4198 . 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 242 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 set class
Syntax hints:    <-> wb 178    \/ wo 359    = wceq 1653   A.wral 2707   [_csb 3253    i^i cin 3321   (/)c0 3630  Disj wdisj 4184
This theorem is referenced by:  disji2  4201  disjprg  4210  disjxiun  4211  disjxun  4212  iundisj2  19445  disji2f  24021  disjpreima  24028  disjxpin  24030  iundisj2f  24032  iundisj2fi  24155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-in 3329  df-nul 3631  df-disj 4185
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