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Theorem difeq 27830
Description: Rewriting an equation with set difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
Assertion
Ref Expression
difeq  |-  ( ( A  \  B )  =  C  <->  ( ( C  i^i  B )  =  (/)  /\  ( C  u.  B )  =  ( A  u.  B ) ) )

Proof of Theorem difeq
StepHypRef Expression
1 incom 3632 . . . . 5  |-  ( B  i^i  ( A  \  B ) )  =  ( ( A  \  B )  i^i  B
)
2 disjdif 3844 . . . . 5  |-  ( B  i^i  ( A  \  B ) )  =  (/)
31, 2eqtr3i 2433 . . . 4  |-  ( ( A  \  B )  i^i  B )  =  (/)
4 ineq1 3634 . . . 4  |-  ( ( A  \  B )  =  C  ->  (
( A  \  B
)  i^i  B )  =  ( C  i^i  B ) )
53, 4syl5reqr 2458 . . 3  |-  ( ( A  \  B )  =  C  ->  ( C  i^i  B )  =  (/) )
6 undif1 3847 . . . 4  |-  ( ( A  \  B )  u.  B )  =  ( A  u.  B
)
7 uneq1 3590 . . . 4  |-  ( ( A  \  B )  =  C  ->  (
( A  \  B
)  u.  B )  =  ( C  u.  B ) )
86, 7syl5reqr 2458 . . 3  |-  ( ( A  \  B )  =  C  ->  ( C  u.  B )  =  ( A  u.  B ) )
95, 8jca 530 . 2  |-  ( ( A  \  B )  =  C  ->  (
( C  i^i  B
)  =  (/)  /\  ( C  u.  B )  =  ( A  u.  B ) ) )
10 simpl 455 . . . 4  |-  ( ( ( C  i^i  B
)  =  (/)  /\  ( C  u.  B )  =  ( A  u.  B ) )  -> 
( C  i^i  B
)  =  (/) )
11 disj3 3814 . . . . 5  |-  ( ( C  i^i  B )  =  (/)  <->  C  =  ( C  \  B ) )
12 eqcom 2411 . . . . 5  |-  ( C  =  ( C  \  B )  <->  ( C  \  B )  =  C )
1311, 12bitri 249 . . . 4  |-  ( ( C  i^i  B )  =  (/)  <->  ( C  \  B )  =  C )
1410, 13sylib 196 . . 3  |-  ( ( ( C  i^i  B
)  =  (/)  /\  ( C  u.  B )  =  ( A  u.  B ) )  -> 
( C  \  B
)  =  C )
15 difeq1 3554 . . . . . 6  |-  ( ( C  u.  B )  =  ( A  u.  B )  ->  (
( C  u.  B
)  \  B )  =  ( ( A  u.  B )  \  B ) )
16 difun2 3851 . . . . . 6  |-  ( ( C  u.  B ) 
\  B )  =  ( C  \  B
)
17 difun2 3851 . . . . . 6  |-  ( ( A  u.  B ) 
\  B )  =  ( A  \  B
)
1815, 16, 173eqtr3g 2466 . . . . 5  |-  ( ( C  u.  B )  =  ( A  u.  B )  ->  ( C  \  B )  =  ( A  \  B
) )
1918eqeq1d 2404 . . . 4  |-  ( ( C  u.  B )  =  ( A  u.  B )  ->  (
( C  \  B
)  =  C  <->  ( A  \  B )  =  C ) )
2019adantl 464 . . 3  |-  ( ( ( C  i^i  B
)  =  (/)  /\  ( C  u.  B )  =  ( A  u.  B ) )  -> 
( ( C  \  B )  =  C  <-> 
( A  \  B
)  =  C ) )
2114, 20mpbid 210 . 2  |-  ( ( ( C  i^i  B
)  =  (/)  /\  ( C  u.  B )  =  ( A  u.  B ) )  -> 
( A  \  B
)  =  C )
229, 21impbii 187 1  |-  ( ( A  \  B )  =  C  <->  ( ( C  i^i  B )  =  (/)  /\  ( C  u.  B )  =  ( A  u.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    \ cdif 3411    u. cun 3412    i^i cin 3413   (/)c0 3738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739
This theorem is referenced by:  difioo  28041
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