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Theorem sossfld 5459
 Description: The base set of a strict order is contained in the field of the relation, except possibly for one element (note that ). (Contributed by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
sossfld

Proof of Theorem sossfld
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eldifsn 4155 . . 3
2 sotrieq 4832 . . . . . . 7
32necon2abid 2711 . . . . . 6
43anass1rs 807 . . . . 5
5 breldmg 5213 . . . . . . . . . 10
653expia 1198 . . . . . . . . 9
76adantll 713 . . . . . . . 8
87an32s 804 . . . . . . 7
9 brelrng 5237 . . . . . . . . 9
1093expia 1198 . . . . . . . 8
1110adantll 713 . . . . . . 7
128, 11orim12d 838 . . . . . 6
13 elun 3644 . . . . . 6
1412, 13syl6ibr 227 . . . . 5
154, 14sylbird 235 . . . 4
1615expimpd 603 . . 3
171, 16syl5bi 217 . 2
1817ssrdv 3509 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  \/wo 368  /\wa 369  e.wcel 1818  =/=wne 2652  \cdif 3472  u.cun 3473  C_wss 3475  {csn 4029   class class class wbr 4452  Orwor 4804  domcdm 5004  rancrn 5005 This theorem is referenced by:  sofld  5460  soex  6743 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-po 4805  df-so 4806  df-cnv 5012  df-dm 5014  df-rn 5015
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