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Theorem uneqin 3216
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 3025 . . . 4  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  ( A  u.  B )  C_  ( A  i^i  B ) )
2 unss 3145 . . . . 5  |-  ( ( A  C_  ( A  i^i  B )  /\  B  C_  ( A  i^i  B
) )  <->  ( A  u.  B )  C_  ( A  i^i  B ) )
3 ssin 3187 . . . . . . 7  |-  ( ( A  C_  A  /\  A  C_  B )  <->  A  C_  ( A  i^i  B ) )
4 sstr 2981 . . . . . . 7  |-  ( ( A  C_  A  /\  A  C_  B )  ->  A  C_  B )
53, 4sylbir 129 . . . . . 6  |-  ( A 
C_  ( A  i^i  B )  ->  A  C_  B
)
6 ssin 3187 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  <->  B  C_  ( A  i^i  B ) )
7 simpl 106 . . . . . . 7  |-  ( ( B  C_  A  /\  B  C_  B )  ->  B  C_  A )
86, 7sylbir 129 . . . . . 6  |-  ( B 
C_  ( A  i^i  B )  ->  B  C_  A
)
95, 8anim12i 325 . . . . 5  |-  ( ( A  C_  ( A  i^i  B )  /\  B  C_  ( A  i^i  B
) )  ->  ( A  C_  B  /\  B  C_  A ) )
102, 9sylbir 129 . . . 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 2988 . . 3  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
1311, 12sylibr 141 . 2  |-  ( ( A  u.  B )  =  ( A  i^i  B )  ->  A  =  B )
14 unidm 3114 . . . 4  |-  ( A  u.  A )  =  A
15 inidm 3174 . . . 4  |-  ( A  i^i  A )  =  A
1614, 15eqtr4i 2079 . . 3  |-  ( A  u.  A )  =  ( A  i^i  A
)
17 uneq2 3119 . . 3  |-  ( A  =  B  ->  ( A  u.  A )  =  ( A  u.  B ) )
18 ineq2 3160 . . 3  |-  ( A  =  B  ->  ( A  i^i  A )  =  ( A  i^i  B
) )
1916, 17, 183eqtr3a 2112 . 2  |-  ( A  =  B  ->  ( A  u.  B )  =  ( A  i^i  B ) )
2013, 19impbii 121 1  |-  ( ( A  u.  B )  =  ( A  i^i  B )  <->  A  =  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259    u. cun 2943    i^i cin 2944    C_ wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959
This theorem is referenced by: (None)
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