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Theorem uneqin 3662
Description: Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
uneqin  |-  ( ( A  u.  B )  =  ( A  i^i  B )  <->  A  =  B
)

Proof of Theorem uneqin
StepHypRef Expression
1 eqimss 3452 . . . 4  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  ( A  u.  B )  C_  ( A  i^i  B ) )
2 unss 3576 . . . . 5  |-  ( ( A  C_  ( A  i^i  B )  /\  B  C_  ( A  i^i  B
) )  <->  ( A  u.  B )  C_  ( A  i^i  B ) )
3 ssin 3622 . . . . . . 7  |-  ( ( A  C_  A  /\  A  C_  B )  <->  A  C_  ( A  i^i  B ) )
4 sstr 3408 . . . . . . 7  |-  ( ( A  C_  A  /\  A  C_  B )  ->  A  C_  B )
53, 4sylbir 218 . . . . . 6  |-  ( A 
C_  ( A  i^i  B )  ->  A  C_  B
)
6 ssin 3622 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  <->  B  C_  ( A  i^i  B ) )
7 simpl 463 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  ->  B  C_  A )
86, 7sylbir 218 . . . . . 6  |-  ( B 
C_  ( A  i^i  B )  ->  B  C_  A
)
95, 8anim12i 574 . . . . 5  |-  ( ( A  C_  ( A  i^i  B )  /\  B  C_  ( A  i^i  B
) )  ->  ( A  C_  B  /\  B  C_  A ) )
102, 9sylbir 218 . . . 4  |-  ( ( A  u.  B ) 
C_  ( A  i^i  B )  ->  ( A  C_  B  /\  B  C_  A ) )
111, 10syl 17 . . 3  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  ( A  C_  B  /\  B  C_  A ) )
12 eqss 3415 . . 3  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
1311, 12sylibr 217 . 2  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  A  =  B )
14 unidm 3545 . . . 4  |-  ( A  u.  A )  =  A
15 inidm 3609 . . . 4  |-  ( A  i^i  A )  =  A
1614, 15eqtr4i 2477 . . 3  |-  ( A  u.  A )  =  ( A  i^i  A
)
17 uneq2 3550 . . 3  |-  ( A  =  B  ->  ( A  u.  A )  =  ( A  u.  B ) )
18 ineq2 3596 . . 3  |-  ( A  =  B  ->  ( A  i^i  A )  =  ( A  i^i  B
) )
1916, 17, 183eqtr3a 2510 . 2  |-  ( A  =  B  ->  ( A  u.  B )  =  ( A  i^i  B ) )
2013, 19impbii 192 1  |-  ( ( A  u.  B )  =  ( A  i^i  B )  <->  A  =  B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1448    u. cun 3370    i^i cin 3371    C_ wss 3372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-v 3015  df-un 3377  df-in 3379  df-ss 3386
This theorem is referenced by:  uniintsn  4242
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