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Theorem uneqin 3420
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 3230 . . . 4  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  ( A  u.  B )  C_  ( A  i^i  B ) )
2 unss 3349 . . . . 5  |-  ( ( A  C_  ( A  i^i  B )  /\  B  C_  ( A  i^i  B
) )  <->  ( A  u.  B )  C_  ( A  i^i  B ) )
3 ssin 3391 . . . . . . 7  |-  ( ( A  C_  A  /\  A  C_  B )  <->  A  C_  ( A  i^i  B ) )
4 sstr 3187 . . . . . . 7  |-  ( ( A  C_  A  /\  A  C_  B )  ->  A  C_  B )
53, 4sylbir 204 . . . . . 6  |-  ( A 
C_  ( A  i^i  B )  ->  A  C_  B
)
6 ssin 3391 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  <->  B  C_  ( A  i^i  B ) )
7 simpl 443 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  ->  B  C_  A )
86, 7sylbir 204 . . . . . 6  |-  ( B 
C_  ( A  i^i  B )  ->  B  C_  A
)
95, 8anim12i 549 . . . . 5  |-  ( ( A  C_  ( A  i^i  B )  /\  B  C_  ( A  i^i  B
) )  ->  ( A  C_  B  /\  B  C_  A ) )
102, 9sylbir 204 . . . 4  |-  ( ( A  u.  B ) 
C_  ( A  i^i  B )  ->  ( A  C_  B  /\  B  C_  A ) )
111, 10syl 15 . . 3  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  ( A  C_  B  /\  B  C_  A ) )
12 eqss 3194 . . 3  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
1311, 12sylibr 203 . 2  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  A  =  B )
14 unidm 3318 . . . 4  |-  ( A  u.  A )  =  A
15 inidm 3378 . . . 4  |-  ( A  i^i  A )  =  A
1614, 15eqtr4i 2306 . . 3  |-  ( A  u.  A )  =  ( A  i^i  A
)
17 uneq2 3323 . . 3  |-  ( A  =  B  ->  ( A  u.  A )  =  ( A  u.  B ) )
18 ineq2 3364 . . 3  |-  ( A  =  B  ->  ( A  i^i  A )  =  ( A  i^i  B
) )
1916, 17, 183eqtr3a 2339 . 2  |-  ( A  =  B  ->  ( A  u.  B )  =  ( A  i^i  B ) )
2013, 19impbii 180 1  |-  ( ( A  u.  B )  =  ( A  i^i  B )  <->  A  =  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    u. cun 3150    i^i cin 3151    C_ wss 3152
This theorem is referenced by:  uniintsn  3899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166
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