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Theorem nepss 23243
Description: Two classes are inequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)
Assertion
Ref Expression
nepss  |-  ( A  =/=  B  <->  ( ( A  i^i  B )  C.  A  \/  ( A  i^i  B )  C.  B
) )

Proof of Theorem nepss
StepHypRef Expression
1 nne 2416 . . . . . 6  |-  ( -.  ( A  i^i  B
)  =/=  A  <->  ( A  i^i  B )  =  A )
2 neeq1 2420 . . . . . . 7  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  =/=  B  <->  A  =/=  B ) )
32biimprcd 218 . . . . . 6  |-  ( A  =/=  B  ->  (
( A  i^i  B
)  =  A  -> 
( A  i^i  B
)  =/=  B ) )
41, 3syl5bi 210 . . . . 5  |-  ( A  =/=  B  ->  ( -.  ( A  i^i  B
)  =/=  A  -> 
( A  i^i  B
)  =/=  B ) )
54orrd 369 . . . 4  |-  ( A  =/=  B  ->  (
( A  i^i  B
)  =/=  A  \/  ( A  i^i  B )  =/=  B ) )
6 inss1 3296 . . . . . 6  |-  ( A  i^i  B )  C_  A
76jctl 527 . . . . 5  |-  ( ( A  i^i  B )  =/=  A  ->  (
( A  i^i  B
)  C_  A  /\  ( A  i^i  B )  =/=  A ) )
8 inss2 3297 . . . . . 6  |-  ( A  i^i  B )  C_  B
98jctl 527 . . . . 5  |-  ( ( A  i^i  B )  =/=  B  ->  (
( A  i^i  B
)  C_  B  /\  ( A  i^i  B )  =/=  B ) )
107, 9orim12i 504 . . . 4  |-  ( ( ( A  i^i  B
)  =/=  A  \/  ( A  i^i  B )  =/=  B )  -> 
( ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A )  \/  ( ( A  i^i  B )  C_  B  /\  ( A  i^i  B )  =/=  B ) ) )
115, 10syl 17 . . 3  |-  ( A  =/=  B  ->  (
( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A )  \/  ( ( A  i^i  B )  C_  B  /\  ( A  i^i  B )  =/=  B ) ) )
12 inidm 3285 . . . . . . 7  |-  ( A  i^i  A )  =  A
13 ineq2 3272 . . . . . . 7  |-  ( A  =  B  ->  ( A  i^i  A )  =  ( A  i^i  B
) )
1412, 13syl5reqr 2300 . . . . . 6  |-  ( A  =  B  ->  ( A  i^i  B )  =  A )
1514necon3i 2451 . . . . 5  |-  ( ( A  i^i  B )  =/=  A  ->  A  =/=  B )
1615adantl 454 . . . 4  |-  ( ( ( A  i^i  B
)  C_  A  /\  ( A  i^i  B )  =/=  A )  ->  A  =/=  B )
17 ineq1 3271 . . . . . . 7  |-  ( A  =  B  ->  ( A  i^i  B )  =  ( B  i^i  B
) )
18 inidm 3285 . . . . . . 7  |-  ( B  i^i  B )  =  B
1917, 18syl6eq 2301 . . . . . 6  |-  ( A  =  B  ->  ( A  i^i  B )  =  B )
2019necon3i 2451 . . . . 5  |-  ( ( A  i^i  B )  =/=  B  ->  A  =/=  B )
2120adantl 454 . . . 4  |-  ( ( ( A  i^i  B
)  C_  B  /\  ( A  i^i  B )  =/=  B )  ->  A  =/=  B )
2216, 21jaoi 370 . . 3  |-  ( ( ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A )  \/  ( ( A  i^i  B )  C_  B  /\  ( A  i^i  B )  =/=  B ) )  ->  A  =/=  B
)
2311, 22impbii 182 . 2  |-  ( A  =/=  B  <->  ( (
( A  i^i  B
)  C_  A  /\  ( A  i^i  B )  =/=  A )  \/  ( ( A  i^i  B )  C_  B  /\  ( A  i^i  B )  =/=  B ) ) )
24 df-pss 3091 . . 3  |-  ( ( A  i^i  B ) 
C.  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
25 df-pss 3091 . . 3  |-  ( ( A  i^i  B ) 
C.  B  <->  ( ( A  i^i  B )  C_  B  /\  ( A  i^i  B )  =/=  B ) )
2624, 25orbi12i 509 . 2  |-  ( ( ( A  i^i  B
)  C.  A  \/  ( A  i^i  B ) 
C.  B )  <->  ( (
( A  i^i  B
)  C_  A  /\  ( A  i^i  B )  =/=  A )  \/  ( ( A  i^i  B )  C_  B  /\  ( A  i^i  B )  =/=  B ) ) )
2723, 26bitr4i 245 1  |-  ( A  =/=  B  <->  ( ( A  i^i  B )  C.  A  \/  ( A  i^i  B )  C.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    =/= wne 2412    i^i cin 3077    C_ wss 3078    C. wpss 3079
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-v 2729  df-in 3085  df-ss 3089  df-pss 3091
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