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Theorem disjor 4109
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 4096 . 2  |-  (Disj  i  e.  A B  <->  A. x E* i  e.  A x  e.  B )
2 ralcom4 2891 . . 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 376 . . . . . . 7  |-  ( ( i  =  j  \/  ( B  i^i  C
)  =  (/) )  <->  ( ( B  i^i  C )  =  (/)  \/  i  =  j ) )
4 df-or 359 . . . . . . 7  |-  ( ( ( B  i^i  C
)  =  (/)  \/  i  =  j )  <->  ( -.  ( B  i^i  C )  =  (/)  ->  i  =  j ) )
5 neq0 3553 . . . . . . . . . 10  |-  ( -.  ( B  i^i  C
)  =  (/)  <->  E. x  x  e.  ( B  i^i  C ) )
6 elin 3446 . . . . . . . . . . 11  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
76exbii 1587 . . . . . . . . . 10  |-  ( E. x  x  e.  ( B  i^i  C )  <->  E. x ( x  e.  B  /\  x  e.  C ) )
85, 7bitri 240 . . . . . . . . 9  |-  ( -.  ( B  i^i  C
)  =  (/)  <->  E. x
( x  e.  B  /\  x  e.  C
) )
98imbi1i 315 . . . . . . . 8  |-  ( ( -.  ( B  i^i  C )  =  (/)  ->  i  =  j )  <->  ( E. x ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
10 19.23v 1901 . . . . . . . 8  |-  ( A. x ( ( x  e.  B  /\  x  e.  C )  ->  i  =  j )  <->  ( E. x ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
119, 10bitr4i 243 . . . . . . 7  |-  ( ( -.  ( B  i^i  C )  =  (/)  ->  i  =  j )  <->  A. x
( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
123, 4, 113bitri 262 . . . . . 6  |-  ( ( i  =  j  \/  ( B  i^i  C
)  =  (/) )  <->  A. x
( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
1312ralbii 2652 . . . . 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 2891 . . . . 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 240 . . . 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 2652 . . 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 2433 . . . . 5  |-  ( i  =  j  ->  (
x  e.  B  <->  x  e.  C ) )
1918rmo4 3044 . . . 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 1571 . . 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 268 . 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 243 1  |-  (Disj  i  e.  A B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1545   E.wex 1546    = wceq 1647    e. wcel 1715   A.wral 2628   E*wrmo 2631    i^i cin 3237   (/)c0 3543  Disj wdisj 4095
This theorem is referenced by:  disjmoOLD  4110  disjors  4111  disjxiun  4122  disjxun  4123  qsdisj2  6879  dyadmbl  19170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rmo 2636  df-v 2875  df-dif 3241  df-in 3245  df-nul 3544  df-disj 4096
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