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Theorem disjor 4431
Description: Two ways to say that a collection  B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
Hypothesis
Ref Expression
disjmo.1  |-  ( i  =  j  ->  B  =  C )
Assertion
Ref Expression
disjor  |-  (Disj  i  e.  A  B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
Distinct variable groups:    i, j, A    B, j    C, i
Allowed substitution hints:    B( i)    C( j)

Proof of Theorem disjor
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-disj 4418 . 2  |-  (Disj  i  e.  A  B  <->  A. x E* i  e.  A  x  e.  B )
2 ralcom4 3132 . . 3  |-  ( A. i  e.  A  A. x A. j  e.  A  ( ( x  e.  B  /\  x  e.  C )  ->  i  =  j )  <->  A. x A. i  e.  A  A. j  e.  A  ( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
3 orcom 387 . . . . . . 7  |-  ( ( i  =  j  \/  ( B  i^i  C
)  =  (/) )  <->  ( ( B  i^i  C )  =  (/)  \/  i  =  j ) )
4 df-or 370 . . . . . . 7  |-  ( ( ( B  i^i  C
)  =  (/)  \/  i  =  j )  <->  ( -.  ( B  i^i  C )  =  (/)  ->  i  =  j ) )
5 neq0 3795 . . . . . . . . . 10  |-  ( -.  ( B  i^i  C
)  =  (/)  <->  E. x  x  e.  ( B  i^i  C ) )
6 elin 3687 . . . . . . . . . . 11  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
76exbii 1644 . . . . . . . . . 10  |-  ( E. x  x  e.  ( B  i^i  C )  <->  E. x ( x  e.  B  /\  x  e.  C ) )
85, 7bitri 249 . . . . . . . . 9  |-  ( -.  ( B  i^i  C
)  =  (/)  <->  E. x
( x  e.  B  /\  x  e.  C
) )
98imbi1i 325 . . . . . . . 8  |-  ( ( -.  ( B  i^i  C )  =  (/)  ->  i  =  j )  <->  ( E. x ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
10 19.23v 1932 . . . . . . . 8  |-  ( A. x ( ( x  e.  B  /\  x  e.  C )  ->  i  =  j )  <->  ( E. x ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
119, 10bitr4i 252 . . . . . . 7  |-  ( ( -.  ( B  i^i  C )  =  (/)  ->  i  =  j )  <->  A. x
( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
123, 4, 113bitri 271 . . . . . 6  |-  ( ( i  =  j  \/  ( B  i^i  C
)  =  (/) )  <->  A. x
( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
1312ralbii 2895 . . . . 5  |-  ( A. j  e.  A  (
i  =  j  \/  ( B  i^i  C
)  =  (/) )  <->  A. j  e.  A  A. x
( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
14 ralcom4 3132 . . . . 5  |-  ( A. j  e.  A  A. x ( ( x  e.  B  /\  x  e.  C )  ->  i  =  j )  <->  A. x A. j  e.  A  ( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
1513, 14bitri 249 . . . 4  |-  ( A. j  e.  A  (
i  =  j  \/  ( B  i^i  C
)  =  (/) )  <->  A. x A. j  e.  A  ( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
1615ralbii 2895 . . 3  |-  ( A. i  e.  A  A. j  e.  A  (
i  =  j  \/  ( B  i^i  C
)  =  (/) )  <->  A. i  e.  A  A. x A. j  e.  A  ( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
17 disjmo.1 . . . . . 6  |-  ( i  =  j  ->  B  =  C )
1817eleq2d 2537 . . . . 5  |-  ( i  =  j  ->  (
x  e.  B  <->  x  e.  C ) )
1918rmo4 3296 . . . 4  |-  ( E* i  e.  A  x  e.  B  <->  A. i  e.  A  A. j  e.  A  ( (
x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
2019albii 1620 . . 3  |-  ( A. x E* i  e.  A  x  e.  B  <->  A. x A. i  e.  A  A. j  e.  A  ( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
212, 16, 203bitr4i 277 . 2  |-  ( A. i  e.  A  A. j  e.  A  (
i  =  j  \/  ( B  i^i  C
)  =  (/) )  <->  A. x E* i  e.  A  x  e.  B )
221, 21bitr4i 252 1  |-  (Disj  i  e.  A  B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2814   E*wrmo 2817    i^i cin 3475   (/)c0 3785  Disj wdisj 4417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rmo 2822  df-v 3115  df-dif 3479  df-in 3483  df-nul 3786  df-disj 4418
This theorem is referenced by:  disjmoOLD  4432  disjors  4433  disjxiun  4444  disjxun  4445  otsndisj  4752  otiunsndisj  4753  qsdisj2  7389  cshwsdisj  14440  dyadmbl  21760  2spotdisj  24754  2spotiundisj  24755  2spotmdisj  24761  numclwwlkdisj  24773  disjnf  27122  disjorsf  27130  mblfinlem2  29645  otiunsndisjX  31784
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