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Theorem disjors 4025
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 2432 . . 3  |-  F/_ i B
2 nfcsb1v 3126 . . 3  |-  F/_ x [_ i  /  x ]_ B
3 csbeq1a 3102 . . 3  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
41, 2, 3cbvdisj 4019 . 2  |-  (Disj  x  e.  A B  <-> Disj  i  e.  A [_ i  /  x ]_ B )
5 csbeq1 3097 . . 3  |-  ( i  =  j  ->  [_ i  /  x ]_ B  = 
[_ j  /  x ]_ B )
65disjor 4023 . 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 240 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 176    \/ wo 357    = wceq 1632   A.wral 2556   [_csb 3094    i^i cin 3164   (/)c0 3468  Disj wdisj 4009
This theorem is referenced by:  disji2  4026  disjprg  4035  disjxiun  4036  disjxun  4037  iundisj2  18922  disji2f  23369  disjpreima  23376  iundisj2fi  23379  iundisj2f  23381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-in 3172  df-nul 3469  df-disj 4010
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