Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjorf Unicode version

Theorem disjorf 23974
Description: Two ways to say that a collection  B ( i ) for  i  e.  A is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Hypotheses
Ref Expression
disjorf.1  |-  F/_ i A
disjorf.2  |-  F/_ j A
disjorf.3  |-  ( i  =  j  ->  B  =  C )
Assertion
Ref Expression
disjorf  |-  (Disj  i  e.  A B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
Distinct variable groups:    i, j    B, j    C, i
Allowed substitution hints:    A( i, j)    B( i)    C( j)

Proof of Theorem disjorf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-disj 4143 . 2  |-  (Disj  i  e.  A B  <->  A. x E* i  e.  A x  e.  B )
2 ralcom4 2934 . . 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 377 . . . . . . 7  |-  ( ( i  =  j  \/  ( B  i^i  C
)  =  (/) )  <->  ( ( B  i^i  C )  =  (/)  \/  i  =  j ) )
4 df-or 360 . . . . . . 7  |-  ( ( ( B  i^i  C
)  =  (/)  \/  i  =  j )  <->  ( -.  ( B  i^i  C )  =  (/)  ->  i  =  j ) )
5 neq0 3598 . . . . . . . . . 10  |-  ( -.  ( B  i^i  C
)  =  (/)  <->  E. x  x  e.  ( B  i^i  C ) )
6 elin 3490 . . . . . . . . . . 11  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
76exbii 1589 . . . . . . . . . 10  |-  ( E. x  x  e.  ( B  i^i  C )  <->  E. x ( x  e.  B  /\  x  e.  C ) )
85, 7bitri 241 . . . . . . . . 9  |-  ( -.  ( B  i^i  C
)  =  (/)  <->  E. x
( x  e.  B  /\  x  e.  C
) )
98imbi1i 316 . . . . . . . 8  |-  ( ( -.  ( B  i^i  C )  =  (/)  ->  i  =  j )  <->  ( E. x ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
10 19.23v 1910 . . . . . . . 8  |-  ( A. x ( ( x  e.  B  /\  x  e.  C )  ->  i  =  j )  <->  ( E. x ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
119, 10bitr4i 244 . . . . . . 7  |-  ( ( -.  ( B  i^i  C )  =  (/)  ->  i  =  j )  <->  A. x
( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
123, 4, 113bitri 263 . . . . . 6  |-  ( ( i  =  j  \/  ( B  i^i  C
)  =  (/) )  <->  A. x
( ( x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
1312ralbii 2690 . . . . 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 2934 . . . . 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 241 . . . 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 2690 . . 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 disjorf.1 . . . . 5  |-  F/_ i A
18 disjorf.2 . . . . 5  |-  F/_ j A
19 nfv 1626 . . . . 5  |-  F/ i  x  e.  C
20 disjorf.3 . . . . . 6  |-  ( i  =  j  ->  B  =  C )
2120eleq2d 2471 . . . . 5  |-  ( i  =  j  ->  (
x  e.  B  <->  x  e.  C ) )
2217, 18, 19, 21rmo4f 23937 . . . 4  |-  ( E* i  e.  A x  e.  B  <->  A. i  e.  A  A. j  e.  A  ( (
x  e.  B  /\  x  e.  C )  ->  i  =  j ) )
2322albii 1572 . . 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 ) )
242, 16, 233bitr4i 269 . 2  |-  ( A. i  e.  A  A. j  e.  A  (
i  =  j  \/  ( B  i^i  C
)  =  (/) )  <->  A. x E* i  e.  A x  e.  B )
251, 24bitr4i 244 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 177    \/ wo 358    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721   F/_wnfc 2527   A.wral 2666   E*wrmo 2669    i^i cin 3279   (/)c0 3588  Disj wdisj 4142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rmo 2674  df-v 2918  df-dif 3283  df-in 3287  df-nul 3589  df-disj 4143
  Copyright terms: Public domain W3C validator