Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nepss Structured version   Unicode version

Theorem nepss 28920
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 2668 . . . . . 6  |-  ( -.  ( A  i^i  B
)  =/=  A  <->  ( A  i^i  B )  =  A )
2 neeq1 2748 . . . . . . 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 3723 . . . . . 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 3724 . . . . . 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 3712 . . . . . . 7  |-  ( A  i^i  A )  =  A
13 ineq2 3699 . . . . . . 7  |-  ( A  =  B  ->  ( A  i^i  A )  =  ( A  i^i  B
) )
1412, 13syl5reqr 2523 . . . . . 6  |-  ( A  =  B  ->  ( A  i^i  B )  =  A )
1514necon3i 2707 . . . . 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 3698 . . . . . . 7  |-  ( A  =  B  ->  ( A  i^i  B )  =  ( B  i^i  B
) )
18 inidm 3712 . . . . . . 7  |-  ( B  i^i  B )  =  B
1917, 18syl6eq 2524 . . . . . 6  |-  ( A  =  B  ->  ( A  i^i  B )  =  B )
2019necon3i 2707 . . . . 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 3497 . . 3  |-  ( ( A  i^i  B ) 
C.  A  <->  ( ( A  i^i  B )  C_  A  /\  ( A  i^i  B )  =/=  A ) )
25 df-pss 3497 . . 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 1379    =/= wne 2662    i^i cin 3480    C_ wss 3481    C. wpss 3482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3120  df-in 3488  df-ss 3495  df-pss 3497
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator