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Theorem disjdifprg2 23971
Description: A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
disjdifprg2  |-  ( A  e.  V  -> Disj  x  e. 
{ ( A  \  B ) ,  ( A  i^i  B ) } x )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem disjdifprg2
StepHypRef Expression
1 inex1g 4306 . . 3  |-  ( A  e.  V  ->  ( A  i^i  B )  e. 
_V )
2 elex 2924 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
3 disjdifprg 23970 . . 3  |-  ( ( ( A  i^i  B
)  e.  _V  /\  A  e.  _V )  -> Disj  x  e.  { ( A  \  ( A  i^i  B ) ) ,  ( A  i^i  B ) } x )
41, 2, 3syl2anc 643 . 2  |-  ( A  e.  V  -> Disj  x  e. 
{ ( A  \ 
( A  i^i  B
) ) ,  ( A  i^i  B ) } x )
5 difin 3538 . . . . 5  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
65preq1i 3846 . . . 4  |-  { ( A  \  ( A  i^i  B ) ) ,  ( A  i^i  B ) }  =  {
( A  \  B
) ,  ( A  i^i  B ) }
76a1i 11 . . 3  |-  ( A  e.  V  ->  { ( A  \  ( A  i^i  B ) ) ,  ( A  i^i  B ) }  =  {
( A  \  B
) ,  ( A  i^i  B ) } )
87disjeq1d 4150 . 2  |-  ( A  e.  V  ->  (Disj  x  e.  { ( A 
\  ( A  i^i  B ) ) ,  ( A  i^i  B ) } x  <-> Disj  x  e.  {
( A  \  B
) ,  ( A  i^i  B ) } x ) )
94, 8mpbid 202 1  |-  ( A  e.  V  -> Disj  x  e. 
{ ( A  \  B ) ,  ( A  i^i  B ) } x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    i^i cin 3279   {cpr 3775  Disj wdisj 4142
This theorem is referenced by:  measxun2  24517
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  ax-sep 4290  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  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-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-sn 3780  df-pr 3781  df-disj 4143
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