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Theorem ordintdif 4572
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 3630 . . 3  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
21necon3bbii 2582 . 2  |-  ( -.  A  C_  B  <->  ( A  \  B )  =/=  (/) )
3 dfdif2 3273 . . . 4  |-  ( A 
\  B )  =  { x  e.  A  |  -.  x  e.  B }
43inteqi 3997 . . 3  |-  |^| ( A  \  B )  = 
|^| { x  e.  A  |  -.  x  e.  B }
5 ordtri1 4556 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
65con2bid 320 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  <->  -.  A  C_  B
) )
7 ordelord 4545 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
8 ordtri1 4556 . . . . . . . . . . . . 13  |-  ( ( Ord  B  /\  Ord  x )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
98ancoms 440 . . . . . . . . . . . 12  |-  ( ( Ord  x  /\  Ord  B )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
107, 9sylan 458 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  A )  /\  Ord  B )  -> 
( B  C_  x  <->  -.  x  e.  B ) )
1110an32s 780 . . . . . . . . . 10  |-  ( ( ( Ord  A  /\  Ord  B )  /\  x  e.  A )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
1211bicomd 193 . . . . . . . . 9  |-  ( ( ( Ord  A  /\  Ord  B )  /\  x  e.  A )  ->  ( -.  x  e.  B  <->  B 
C_  x ) )
1312rabbidva 2891 . . . . . . . 8  |-  ( ( Ord  A  /\  Ord  B )  ->  { x  e.  A  |  -.  x  e.  B }  =  { x  e.  A  |  B  C_  x }
)
1413inteqd 3998 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  |^| { x  e.  A  |  B  C_  x } )
15 intmin 4013 . . . . . . 7  |-  ( B  e.  A  ->  |^| { x  e.  A  |  B  C_  x }  =  B )
1614, 15sylan9eq 2440 . . . . . 6  |-  ( ( ( Ord  A  /\  Ord  B )  /\  B  e.  A )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B )
1716ex 424 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B ) )
186, 17sylbird 227 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  A  C_  B  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B ) )
19183impia 1150 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  -.  A  C_  B )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B )
204, 19syl5req 2433 . 2  |-  ( ( Ord  A  /\  Ord  B  /\  -.  A  C_  B )  ->  B  =  |^| ( A  \  B ) )
212, 20syl3an3br 1225 1  |-  ( ( Ord  A  /\  Ord  B  /\  ( A  \  B )  =/=  (/) )  ->  B  =  |^| ( A 
\  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   {crab 2654    \ cdif 3261    C_ wss 3264   (/)c0 3572   |^|cint 3993   Ord word 4522
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-int 3994  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526
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