MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordintdif Structured version   Unicode version

Theorem ordintdif 5491
Description: If  B is smaller than  A, then it equals the intersection of the difference. Exercise 11 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)
Assertion
Ref Expression
ordintdif  |-  ( ( Ord  A  /\  Ord  B  /\  ( A  \  B )  =/=  (/) )  ->  B  =  |^| ( A 
\  B ) )

Proof of Theorem ordintdif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssdif0 3853 . . 3  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
21necon3bbii 2681 . 2  |-  ( -.  A  C_  B  <->  ( A  \  B )  =/=  (/) )
3 dfdif2 3445 . . . 4  |-  ( A 
\  B )  =  { x  e.  A  |  -.  x  e.  B }
43inteqi 4259 . . 3  |-  |^| ( A  \  B )  = 
|^| { x  e.  A  |  -.  x  e.  B }
5 ordtri1 5475 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
65con2bid 330 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  <->  -.  A  C_  B
) )
7 ordelord 5464 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
8 ordtri1 5475 . . . . . . . . . . . . 13  |-  ( ( Ord  B  /\  Ord  x )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
98ancoms 454 . . . . . . . . . . . 12  |-  ( ( Ord  x  /\  Ord  B )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
107, 9sylan 473 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  A )  /\  Ord  B )  -> 
( B  C_  x  <->  -.  x  e.  B ) )
1110an32s 811 . . . . . . . . . 10  |-  ( ( ( Ord  A  /\  Ord  B )  /\  x  e.  A )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
1211bicomd 204 . . . . . . . . 9  |-  ( ( ( Ord  A  /\  Ord  B )  /\  x  e.  A )  ->  ( -.  x  e.  B  <->  B 
C_  x ) )
1312rabbidva 3070 . . . . . . . 8  |-  ( ( Ord  A  /\  Ord  B )  ->  { x  e.  A  |  -.  x  e.  B }  =  { x  e.  A  |  B  C_  x }
)
1413inteqd 4260 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  |^| { x  e.  A  |  B  C_  x } )
15 intmin 4275 . . . . . . 7  |-  ( B  e.  A  ->  |^| { x  e.  A  |  B  C_  x }  =  B )
1614, 15sylan9eq 2483 . . . . . 6  |-  ( ( ( Ord  A  /\  Ord  B )  /\  B  e.  A )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B )
1716ex 435 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B ) )
186, 17sylbird 238 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  A  C_  B  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B ) )
19183impia 1202 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  -.  A  C_  B )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B )
204, 19syl5req 2476 . 2  |-  ( ( Ord  A  /\  Ord  B  /\  -.  A  C_  B )  ->  B  =  |^| ( A  \  B ) )
212, 20syl3an3br 1305 1  |-  ( ( Ord  A  /\  Ord  B  /\  ( A  \  B )  =/=  (/) )  ->  B  =  |^| ( A 
\  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   {crab 2775    \ cdif 3433    C_ wss 3436   (/)c0 3761   |^|cint 4255   Ord word 5441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-int 4256  df-br 4424  df-opab 4483  df-tr 4519  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator