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Theorem sossfld 5120
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 3749 . . 3  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
2 sotrieq 4341 . . . . . . 7  |-  ( ( R  Or  A  /\  ( x  e.  A  /\  B  e.  A
) )  ->  (
x  =  B  <->  -.  (
x R B  \/  B R x ) ) )
32necon2abid 2503 . . . . . 6  |-  ( ( R  Or  A  /\  ( x  e.  A  /\  B  e.  A
) )  ->  (
( x R B  \/  B R x )  <->  x  =/=  B
) )
43anass1rs 782 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
( x R B  \/  B R x )  <->  x  =/=  B
) )
5 breldmg 4884 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  A  /\  x R B )  ->  x  e.  dom  R )
653expia 1153 . . . . . . . . 9  |-  ( ( x  e.  A  /\  B  e.  A )  ->  ( x R B  ->  x  e.  dom  R ) )
76adantll 694 . . . . . . . 8  |-  ( ( ( R  Or  A  /\  x  e.  A
)  /\  B  e.  A )  ->  (
x R B  ->  x  e.  dom  R ) )
87an32s 779 . . . . . . 7  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
x R B  ->  x  e.  dom  R ) )
9 brelrng 4908 . . . . . . . . 9  |-  ( ( B  e.  A  /\  x  e.  A  /\  B R x )  ->  x  e.  ran  R )
1093expia 1153 . . . . . . . 8  |-  ( ( B  e.  A  /\  x  e.  A )  ->  ( B R x  ->  x  e.  ran  R ) )
1110adantll 694 . . . . . . 7  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  ( B R x  ->  x  e.  ran  R ) )
128, 11orim12d 811 . . . . . 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 3316 . . . . . 6  |-  ( x  e.  ( dom  R  u.  ran  R )  <->  ( x  e.  dom  R  \/  x  e.  ran  R ) )
1412, 13syl6ibr 218 . . . . 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 226 . . . 4  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
x  =/=  B  ->  x  e.  ( dom  R  u.  ran  R ) ) )
1615expimpd 586 . . 3  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( ( x  e.  A  /\  x  =/= 
B )  ->  x  e.  ( dom  R  u.  ran  R ) ) )
171, 16syl5bi 208 . 2  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( x  e.  ( A  \  { B } )  ->  x  e.  ( dom  R  u.  ran  R ) ) )
1817ssrdv 3185 1  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( A  \  { B } )  C_  ( dom  R  u.  ran  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    e. wcel 1684    =/= wne 2446    \ cdif 3149    u. cun 3150    C_ wss 3152   {csn 3640   class class class wbr 4023    Or wor 4313   dom cdm 4689   ran crn 4690
This theorem is referenced by:  sofld  5121  soex  5122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-cnv 4697  df-dm 4699  df-rn 4700
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