MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sossfld Structured version   Unicode version

Theorem sossfld 5280
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 3995 . . 3  |-  ( x  e.  ( A  \  { B } )  <->  ( x  e.  A  /\  x  =/=  B ) )
2 sotrieq 4663 . . . . . . 7  |-  ( ( R  Or  A  /\  ( x  e.  A  /\  B  e.  A
) )  ->  (
x  =  B  <->  -.  (
x R B  \/  B R x ) ) )
32necon2abid 2663 . . . . . 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 5040 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  A  /\  x R B )  ->  x  e.  dom  R )
653expia 1189 . . . . . . . . 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 5064 . . . . . . . . 9  |-  ( ( B  e.  A  /\  x  e.  A  /\  B R x )  ->  x  e.  ran  R )
1093expia 1189 . . . . . . . 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 834 . . . . . 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 3492 . . . . . 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 3357 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 1756    =/= wne 2601    \ cdif 3320    u. cun 3321    C_ wss 3323   {csn 3872   class class class wbr 4287    Or wor 4635   dom cdm 4835   ran crn 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-po 4636  df-so 4637  df-cnv 4843  df-dm 4845  df-rn 4846
This theorem is referenced by:  sofld  5281  soex  6516
  Copyright terms: Public domain W3C validator