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Theorem nepss 27511
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 2650 . . . . . 6  |-  ( -.  ( A  i^i  B
)  =/=  A  <->  ( A  i^i  B )  =  A )
2 neeq1 2729 . . . . . . 7  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  =/=  B  <->  A  =/=  B ) )
32biimprcd 225 . . . . . 6  |-  ( A  =/=  B  ->  (
( A  i^i  B
)  =  A  -> 
( A  i^i  B
)  =/=  B ) )
41, 3syl5bi 217 . . . . 5  |-  ( A  =/=  B  ->  ( -.  ( A  i^i  B
)  =/=  A  -> 
( A  i^i  B
)  =/=  B ) )
54orrd 378 . . . 4  |-  ( A  =/=  B  ->  (
( A  i^i  B
)  =/=  A  \/  ( A  i^i  B )  =/=  B ) )
6 inss1 3671 . . . . . 6  |-  ( A  i^i  B )  C_  A
76jctl 541 . . . . 5  |-  ( ( A  i^i  B )  =/=  A  ->  (
( A  i^i  B
)  C_  A  /\  ( A  i^i  B )  =/=  A ) )
8 inss2 3672 . . . . . 6  |-  ( A  i^i  B )  C_  B
98jctl 541 . . . . 5  |-  ( ( A  i^i  B )  =/=  B  ->  (
( A  i^i  B
)  C_  B  /\  ( A  i^i  B )  =/=  B ) )
107, 9orim12i 516 . . . 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 16 . . 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 3660 . . . . . . 7  |-  ( A  i^i  A )  =  A
13 ineq2 3647 . . . . . . 7  |-  ( A  =  B  ->  ( A  i^i  A )  =  ( A  i^i  B
) )
1412, 13syl5reqr 2507 . . . . . 6  |-  ( A  =  B  ->  ( A  i^i  B )  =  A )
1514necon3i 2688 . . . . 5  |-  ( ( A  i^i  B )  =/=  A  ->  A  =/=  B )
1615adantl 466 . . . 4  |-  ( ( ( A  i^i  B
)  C_  A  /\  ( A  i^i  B )  =/=  A )  ->  A  =/=  B )
17 ineq1 3646 . . . . . . 7  |-  ( A  =  B  ->  ( A  i^i  B )  =  ( B  i^i  B
) )
18 inidm 3660 . . . . . . 7  |-  ( B  i^i  B )  =  B
1917, 18syl6eq 2508 . . . . . 6  |-  ( A  =  B  ->  ( A  i^i  B )  =  B )
2019necon3i 2688 . . . . 5  |-  ( ( A  i^i  B )  =/=  B  ->  A  =/=  B )
2120adantl 466 . . . 4  |-  ( ( ( A  i^i  B
)  C_  B  /\  ( A  i^i  B )  =/=  B )  ->  A  =/=  B )
2216, 21jaoi 379 . . 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 188 . 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 3445 . . 3  |-  ( ( A  i^i  B ) 
C.  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
25 df-pss 3445 . . 3  |-  ( ( A  i^i  B ) 
C.  B  <->  ( ( A  i^i  B )  C_  B  /\  ( A  i^i  B )  =/=  B ) )
2624, 25orbi12i 521 . 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 252 1  |-  ( A  =/=  B  <->  ( ( A  i^i  B )  C.  A  \/  ( A  i^i  B )  C.  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    =/= wne 2644    i^i cin 3428    C_ wss 3429    C. wpss 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3073  df-in 3436  df-ss 3443  df-pss 3445
This theorem is referenced by: (None)
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