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Theorem disjors 4009
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 2419 . . 3  |-  F/_ i B
2 nfcsb1v 3113 . . 3  |-  F/_ x [_ i  /  x ]_ B
3 csbeq1a 3089 . . 3  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
41, 2, 3cbvdisj 4003 . 2  |-  (Disj  x  e.  A B  <-> Disj  i  e.  A [_ i  /  x ]_ B )
5 csbeq1 3084 . . 3  |-  ( i  =  j  ->  [_ i  /  x ]_ B  = 
[_ j  /  x ]_ B )
65disjor 4007 . 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 1623   A.wral 2543   [_csb 3081    i^i cin 3151   (/)c0 3455  Disj wdisj 3993
This theorem is referenced by:  disji2  4010  disjprg  4019  disjxiun  4020  disjxun  4021  iundisj2  18906  disji2f  23354  disjpreima  23361  iundisj2fi  23364  iundisj2f  23366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-in 3159  df-nul 3456  df-disj 3994
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