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Theorem uneqin 3185
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 2994 . . . 4  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  ( A  u.  B )  C_  ( A  i^i  B ) )
2 unss 3114 . . . . 5  |-  ( ( A  C_  ( A  i^i  B )  /\  B  C_  ( A  i^i  B
) )  <->  ( A  u.  B )  C_  ( A  i^i  B ) )
3 ssin 3156 . . . . . . 7  |-  ( ( A  C_  A  /\  A  C_  B )  <->  A  C_  ( A  i^i  B ) )
4 sstr 2950 . . . . . . 7  |-  ( ( A  C_  A  /\  A  C_  B )  ->  A  C_  B )
53, 4sylbir 125 . . . . . 6  |-  ( A 
C_  ( A  i^i  B )  ->  A  C_  B
)
6 ssin 3156 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  <->  B  C_  ( A  i^i  B ) )
7 simpl 102 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  ->  B  C_  A )
86, 7sylbir 125 . . . . . 6  |-  ( B 
C_  ( A  i^i  B )  ->  B  C_  A
)
95, 8anim12i 321 . . . . 5  |-  ( ( A  C_  ( A  i^i  B )  /\  B  C_  ( A  i^i  B
) )  ->  ( A  C_  B  /\  B  C_  A ) )
102, 9sylbir 125 . . . 4  |-  ( ( A  u.  B ) 
C_  ( A  i^i  B )  ->  ( A  C_  B  /\  B  C_  A ) )
111, 10syl 14 . . 3  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  ( A  C_  B  /\  B  C_  A ) )
12 eqss 2957 . . 3  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
1311, 12sylibr 137 . 2  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  A  =  B )
14 unidm 3083 . . . 4  |-  ( A  u.  A )  =  A
15 inidm 3143 . . . 4  |-  ( A  i^i  A )  =  A
1614, 15eqtr4i 2063 . . 3  |-  ( A  u.  A )  =  ( A  i^i  A
)
17 uneq2 3088 . . 3  |-  ( A  =  B  ->  ( A  u.  A )  =  ( A  u.  B ) )
18 ineq2 3129 . . 3  |-  ( A  =  B  ->  ( A  i^i  A )  =  ( A  i^i  B
) )
1916, 17, 183eqtr3a 2096 . 2  |-  ( A  =  B  ->  ( A  u.  B )  =  ( A  i^i  B ) )
2013, 19impbii 117 1  |-  ( ( A  u.  B )  =  ( A  i^i  B )  <->  A  =  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98    = wceq 1243    u. cun 2912    i^i cin 2913    C_ wss 2914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-un 2919  df-in 2921  df-ss 2928
This theorem is referenced by: (None)
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