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Theorem ordintdif 4413
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
StepHypRef Expression
1 ssdif0 3488 . . 3  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
21necon3bbii 2452 . 2  |-  ( -.  A  C_  B  <->  ( A  \  B )  =/=  (/) )
3 dfdif2 3136 . . . 4  |-  ( A 
\  B )  =  { x  e.  A  |  -.  x  e.  B }
43inteqi 3840 . . 3  |-  |^| ( A  \  B )  = 
|^| { x  e.  A  |  -.  x  e.  B }
5 ordtri1 4397 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
65con2bid 321 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  <->  -.  A  C_  B
) )
7 ordelord 4386 . . . . . . . . . . . 12  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
8 ordtri1 4397 . . . . . . . . . . . . 13  |-  ( ( Ord  B  /\  Ord  x )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
98ancoms 441 . . . . . . . . . . . 12  |-  ( ( Ord  x  /\  Ord  B )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
107, 9sylan 459 . . . . . . . . . . 11  |-  ( ( ( Ord  A  /\  x  e.  A )  /\  Ord  B )  -> 
( B  C_  x  <->  -.  x  e.  B ) )
1110an32s 782 . . . . . . . . . 10  |-  ( ( ( Ord  A  /\  Ord  B )  /\  x  e.  A )  ->  ( B  C_  x  <->  -.  x  e.  B ) )
1211bicomd 194 . . . . . . . . 9  |-  ( ( ( Ord  A  /\  Ord  B )  /\  x  e.  A )  ->  ( -.  x  e.  B  <->  B 
C_  x ) )
1312rabbidva 2754 . . . . . . . 8  |-  ( ( Ord  A  /\  Ord  B )  ->  { x  e.  A  |  -.  x  e.  B }  =  { x  e.  A  |  B  C_  x }
)
1413inteqd 3841 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  |^| { x  e.  A  |  B  C_  x } )
15 intmin 3856 . . . . . . 7  |-  ( B  e.  A  ->  |^| { x  e.  A  |  B  C_  x }  =  B )
1614, 15sylan9eq 2310 . . . . . 6  |-  ( ( ( Ord  A  /\  Ord  B )  /\  B  e.  A )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B )
1716ex 425 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B ) )
186, 17sylbird 228 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  A  C_  B  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B ) )
19183impia 1153 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  -.  A  C_  B )  ->  |^| { x  e.  A  |  -.  x  e.  B }  =  B )
204, 19syl5req 2303 . 2  |-  ( ( Ord  A  /\  Ord  B  /\  -.  A  C_  B )  ->  B  =  |^| ( A  \  B ) )
212, 20syl3an3br 1228 1  |-  ( ( Ord  A  /\  Ord  B  /\  ( A  \  B )  =/=  (/) )  ->  B  =  |^| ( A 
\  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   {crab 2522    \ cdif 3124    C_ wss 3127   (/)c0 3430   |^|cint 3836   Ord word 4363
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-int 3837  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367
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