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Theorem sossfld 5305
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 4110 . . 3  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
2 sotrieq 4804 . . . . . . 7  |-  ( ( R  Or  A  /\  ( x  e.  A  /\  B  e.  A
) )  ->  (
x  =  B  <->  -.  (
x R B  \/  B R x ) ) )
32necon2abid 2678 . . . . . 6  |-  ( ( R  Or  A  /\  ( x  e.  A  /\  B  e.  A
) )  ->  (
( x R B  \/  B R x )  <->  x  =/=  B
) )
43anass1rs 821 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
( x R B  \/  B R x )  <->  x  =/=  B
) )
5 breldmg 5062 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  A  /\  x R B )  ->  x  e.  dom  R )
653expia 1217 . . . . . . . . 9  |-  ( ( x  e.  A  /\  B  e.  A )  ->  ( x R B  ->  x  e.  dom  R ) )
76adantll 725 . . . . . . . 8  |-  ( ( ( R  Or  A  /\  x  e.  A
)  /\  B  e.  A )  ->  (
x R B  ->  x  e.  dom  R ) )
87an32s 818 . . . . . . 7  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
x R B  ->  x  e.  dom  R ) )
9 brelrng 5086 . . . . . . . . 9  |-  ( ( B  e.  A  /\  x  e.  A  /\  B R x )  ->  x  e.  ran  R )
1093expia 1217 . . . . . . . 8  |-  ( ( B  e.  A  /\  x  e.  A )  ->  ( B R x  ->  x  e.  ran  R ) )
1110adantll 725 . . . . . . 7  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  ( B R x  ->  x  e.  ran  R ) )
128, 11orim12d 854 . . . . . 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 3586 . . . . . 6  |-  ( x  e.  ( dom  R  u.  ran  R )  <->  ( x  e.  dom  R  \/  x  e.  ran  R ) )
1412, 13syl6ibr 235 . . . . 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 243 . . . 4  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
x  =/=  B  ->  x  e.  ( dom  R  u.  ran  R ) ) )
1615expimpd 612 . . 3  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( ( x  e.  A  /\  x  =/= 
B )  ->  x  e.  ( dom  R  u.  ran  R ) ) )
171, 16syl5bi 225 . 2  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( x  e.  ( A  \  { B } )  ->  x  e.  ( dom  R  u.  ran  R ) ) )
1817ssrdv 3450 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 189    \/ wo 374    /\ wa 375    e. wcel 1898    =/= wne 2633    \ cdif 3413    u. cun 3414    C_ wss 3416   {csn 3980   class class class wbr 4418    Or wor 4776   dom cdm 4856   ran crn 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419  df-opab 4478  df-po 4777  df-so 4778  df-cnv 4864  df-dm 4866  df-rn 4867
This theorem is referenced by:  sofld  5306  soex  6768
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