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Theorem sossfld 5317
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 3927 . . 3  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
2 sotrieq 4530 . . . . . . 7  |-  ( ( R  Or  A  /\  ( x  e.  A  /\  B  e.  A
) )  ->  (
x  =  B  <->  -.  (
x R B  \/  B R x ) ) )
32necon2abid 2661 . . . . . 6  |-  ( ( R  Or  A  /\  ( x  e.  A  /\  B  e.  A
) )  ->  (
( x R B  \/  B R x )  <->  x  =/=  B
) )
43anass1rs 783 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
( x R B  \/  B R x )  <->  x  =/=  B
) )
5 breldmg 5075 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  A  /\  x R B )  ->  x  e.  dom  R )
653expia 1155 . . . . . . . . 9  |-  ( ( x  e.  A  /\  B  e.  A )  ->  ( x R B  ->  x  e.  dom  R ) )
76adantll 695 . . . . . . . 8  |-  ( ( ( R  Or  A  /\  x  e.  A
)  /\  B  e.  A )  ->  (
x R B  ->  x  e.  dom  R ) )
87an32s 780 . . . . . . 7  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
x R B  ->  x  e.  dom  R ) )
9 brelrng 5099 . . . . . . . . 9  |-  ( ( B  e.  A  /\  x  e.  A  /\  B R x )  ->  x  e.  ran  R )
1093expia 1155 . . . . . . . 8  |-  ( ( B  e.  A  /\  x  e.  A )  ->  ( B R x  ->  x  e.  ran  R ) )
1110adantll 695 . . . . . . 7  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  ( B R x  ->  x  e.  ran  R ) )
128, 11orim12d 812 . . . . . 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 3488 . . . . . 6  |-  ( x  e.  ( dom  R  u.  ran  R )  <->  ( x  e.  dom  R  \/  x  e.  ran  R ) )
1412, 13syl6ibr 219 . . . . 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 227 . . . 4  |-  ( ( ( R  Or  A  /\  B  e.  A
)  /\  x  e.  A )  ->  (
x  =/=  B  ->  x  e.  ( dom  R  u.  ran  R ) ) )
1615expimpd 587 . . 3  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( ( x  e.  A  /\  x  =/= 
B )  ->  x  e.  ( dom  R  u.  ran  R ) ) )
171, 16syl5bi 209 . 2  |-  ( ( R  Or  A  /\  B  e.  A )  ->  ( x  e.  ( A  \  { B } )  ->  x  e.  ( dom  R  u.  ran  R ) ) )
1817ssrdv 3354 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 177    \/ wo 358    /\ wa 359    e. wcel 1725    =/= wne 2599    \ cdif 3317    u. cun 3318    C_ wss 3320   {csn 3814   class class class wbr 4212    Or wor 4502   dom cdm 4878   ran crn 4879
This theorem is referenced by:  sofld  5318  soex  5319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-po 4503  df-so 4504  df-cnv 4886  df-dm 4888  df-rn 4889
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