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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 ). (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 3927 . . 3
2 sotrieq 4530 . . . . . . 7
32necon2abid 2661 . . . . . 6
43anass1rs 783 . . . . 5
5 breldmg 5075 . . . . . . . . . 10
653expia 1155 . . . . . . . . 9
76adantll 695 . . . . . . . 8
87an32s 780 . . . . . . 7
9 brelrng 5099 . . . . . . . . 9
1093expia 1155 . . . . . . . 8
1110adantll 695 . . . . . . 7
128, 11orim12d 812 . . . . . 6
13 elun 3488 . . . . . 6
1412, 13syl6ibr 219 . . . . 5
154, 14sylbird 227 . . . 4
1615expimpd 587 . . 3
171, 16syl5bi 209 . 2
1817ssrdv 3354 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wo 358   wa 359   wcel 1725   wne 2599   cdif 3317   cun 3318   wss 3320  csn 3814   class class class wbr 4212   wor 4502   cdm 4878   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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