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Theorem ordintdif 4457
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 3526 . . 3  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
21necon3bbii 2490 . 2  |-  ( -.  A  C_  B  <->  ( A  \  B )  =/=  (/) )
3 dfdif2 3174 . . . 4  |-  ( A 
\  B )  =  { x  e.  A  |  -.  x  e.  B }
43inteqi 3882 . . 3  |-  |^| ( A  \  B )  = 
|^| { x  e.  A  |  -.  x  e.  B }
5 ordtri1 4441 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
65con2bid 319 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  <->  -.  A  C_  B
) )
7 ordelord 4430 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
8 ordtri1 4441 . . . . . . . . . . . . 13  |-  ( ( Ord  B  /\  Ord  x )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
98ancoms 439 . . . . . . . . . . . 12  |-  ( ( Ord  x  /\  Ord  B )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
107, 9sylan 457 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  A )  /\  Ord  B )  -> 
( B  C_  x  <->  -.  x  e.  B ) )
1110an32s 779 . . . . . . . . . 10  |-  ( ( ( Ord  A  /\  Ord  B )  /\  x  e.  A )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
1211bicomd 192 . . . . . . . . 9  |-  ( ( ( Ord  A  /\  Ord  B )  /\  x  e.  A )  ->  ( -.  x  e.  B  <->  B 
C_  x ) )
1312rabbidva 2792 . . . . . . . 8  |-  ( ( Ord  A  /\  Ord  B )  ->  { x  e.  A  |  -.  x  e.  B }  =  { x  e.  A  |  B  C_  x }
)
1413inteqd 3883 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  |^| { x  e.  A  |  B  C_  x } )
15 intmin 3898 . . . . . . 7  |-  ( B  e.  A  ->  |^| { x  e.  A  |  B  C_  x }  =  B )
1614, 15sylan9eq 2348 . . . . . 6  |-  ( ( ( Ord  A  /\  Ord  B )  /\  B  e.  A )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B )
1716ex 423 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B ) )
186, 17sylbird 226 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  A  C_  B  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B ) )
19183impia 1148 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  -.  A  C_  B )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B )
204, 19syl5req 2341 . 2  |-  ( ( Ord  A  /\  Ord  B  /\  -.  A  C_  B )  ->  B  =  |^| ( A  \  B ) )
212, 20syl3an3br 1223 1  |-  ( ( Ord  A  /\  Ord  B  /\  ( A  \  B )  =/=  (/) )  ->  B  =  |^| ( A 
\  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560    \ cdif 3162    C_ wss 3165   (/)c0 3468   |^|cint 3878   Ord word 4407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411
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