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Theorem sossfld 5460
Description: The base set of a strict order is contained in the field of the relation, except possibly for one element (note that  (/)  Or  { B }). (Contributed by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
sossfld  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( A  \  { B } )  C_  ( dom  R  u.  ran  R
) )

Proof of Theorem sossfld
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldifsn 4158 . . 3  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
2 sotrieq 4833 . . . . . . 7  |-  ( ( R  Or  A  /\  ( x  e.  A  /\  B  e.  A
) )  ->  (
x  =  B  <->  -.  (
x R B  \/  B R x ) ) )
32necon2abid 2721 . . . . . 6  |-  ( ( R  Or  A  /\  ( x  e.  A  /\  B  e.  A
) )  ->  (
( x R B  \/  B R x )  <->  x  =/=  B
) )
43anass1rs 805 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
( x R B  \/  B R x )  <->  x  =/=  B
) )
5 breldmg 5214 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  A  /\  x R B )  ->  x  e.  dom  R )
653expia 1198 . . . . . . . . 9  |-  ( ( x  e.  A  /\  B  e.  A )  ->  ( x R B  ->  x  e.  dom  R ) )
76adantll 713 . . . . . . . 8  |-  ( ( ( R  Or  A  /\  x  e.  A
)  /\  B  e.  A )  ->  (
x R B  ->  x  e.  dom  R ) )
87an32s 802 . . . . . . 7  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
x R B  ->  x  e.  dom  R ) )
9 brelrng 5238 . . . . . . . . 9  |-  ( ( B  e.  A  /\  x  e.  A  /\  B R x )  ->  x  e.  ran  R )
1093expia 1198 . . . . . . . 8  |-  ( ( B  e.  A  /\  x  e.  A )  ->  ( B R x  ->  x  e.  ran  R ) )
1110adantll 713 . . . . . . 7  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  ( B R x  ->  x  e.  ran  R ) )
128, 11orim12d 836 . . . . . 6  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
( x R B  \/  B R x )  ->  ( x  e.  dom  R  \/  x  e.  ran  R ) ) )
13 elun 3650 . . . . . 6  |-  ( x  e.  ( dom  R  u.  ran  R )  <->  ( x  e.  dom  R  \/  x  e.  ran  R ) )
1412, 13syl6ibr 227 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
( x R B  \/  B R x )  ->  x  e.  ( dom  R  u.  ran  R ) ) )
154, 14sylbird 235 . . . 4  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
x  =/=  B  ->  x  e.  ( dom  R  u.  ran  R ) ) )
1615expimpd 603 . . 3  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( ( x  e.  A  /\  x  =/= 
B )  ->  x  e.  ( dom  R  u.  ran  R ) ) )
171, 16syl5bi 217 . 2  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( x  e.  ( A  \  { B } )  ->  x  e.  ( dom  R  u.  ran  R ) ) )
1817ssrdv 3515 1  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( A  \  { B } )  C_  ( dom  R  u.  ran  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    e. wcel 1767    =/= wne 2662    \ cdif 3478    u. cun 3479    C_ wss 3481   {csn 4033   class class class wbr 4453    Or wor 4805   dom cdm 5005   ran crn 5006
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-po 4806  df-so 4807  df-cnv 5013  df-dm 5015  df-rn 5016
This theorem is referenced by:  sofld  5461  soex  6738
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