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Theorem nepss 25128
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 2571 . . . . . 6  |-  ( -.  ( A  i^i  B
)  =/=  A  <->  ( A  i^i  B )  =  A )
2 neeq1 2575 . . . . . . 7  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  =/=  B  <->  A  =/=  B ) )
32biimprcd 217 . . . . . 6  |-  ( A  =/=  B  ->  (
( A  i^i  B
)  =  A  -> 
( A  i^i  B
)  =/=  B ) )
41, 3syl5bi 209 . . . . 5  |-  ( A  =/=  B  ->  ( -.  ( A  i^i  B
)  =/=  A  -> 
( A  i^i  B
)  =/=  B ) )
54orrd 368 . . . 4  |-  ( A  =/=  B  ->  (
( A  i^i  B
)  =/=  A  \/  ( A  i^i  B )  =/=  B ) )
6 inss1 3521 . . . . . 6  |-  ( A  i^i  B )  C_  A
76jctl 526 . . . . 5  |-  ( ( A  i^i  B )  =/=  A  ->  (
( A  i^i  B
)  C_  A  /\  ( A  i^i  B )  =/=  A ) )
8 inss2 3522 . . . . . 6  |-  ( A  i^i  B )  C_  B
98jctl 526 . . . . 5  |-  ( ( A  i^i  B )  =/=  B  ->  (
( A  i^i  B
)  C_  B  /\  ( A  i^i  B )  =/=  B ) )
107, 9orim12i 503 . . . 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 3510 . . . . . . 7  |-  ( A  i^i  A )  =  A
13 ineq2 3496 . . . . . . 7  |-  ( A  =  B  ->  ( A  i^i  A )  =  ( A  i^i  B
) )
1412, 13syl5reqr 2451 . . . . . 6  |-  ( A  =  B  ->  ( A  i^i  B )  =  A )
1514necon3i 2606 . . . . 5  |-  ( ( A  i^i  B )  =/=  A  ->  A  =/=  B )
1615adantl 453 . . . 4  |-  ( ( ( A  i^i  B
)  C_  A  /\  ( A  i^i  B )  =/=  A )  ->  A  =/=  B )
17 ineq1 3495 . . . . . . 7  |-  ( A  =  B  ->  ( A  i^i  B )  =  ( B  i^i  B
) )
18 inidm 3510 . . . . . . 7  |-  ( B  i^i  B )  =  B
1917, 18syl6eq 2452 . . . . . 6  |-  ( A  =  B  ->  ( A  i^i  B )  =  B )
2019necon3i 2606 . . . . 5  |-  ( ( A  i^i  B )  =/=  B  ->  A  =/=  B )
2120adantl 453 . . . 4  |-  ( ( ( A  i^i  B
)  C_  B  /\  ( A  i^i  B )  =/=  B )  ->  A  =/=  B )
2216, 21jaoi 369 . . 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 181 . 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 3296 . . 3  |-  ( ( A  i^i  B ) 
C.  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
25 df-pss 3296 . . 3  |-  ( ( A  i^i  B ) 
C.  B  <->  ( ( A  i^i  B )  C_  B  /\  ( A  i^i  B )  =/=  B ) )
2624, 25orbi12i 508 . 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 244 1  |-  ( A  =/=  B  <->  ( ( A  i^i  B )  C.  A  \/  ( A  i^i  B )  C.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    =/= wne 2567    i^i cin 3279    C_ wss 3280    C. wpss 3281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-in 3287  df-ss 3294  df-pss 3296
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