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Theorem sossfld 5484
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 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))

Proof of Theorem sossfld
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eldifsn 4259 . . 3 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
2 sotrieq 4975 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝐵𝐴)) → (𝑥 = 𝐵 ↔ ¬ (𝑥𝑅𝐵𝐵𝑅𝑥)))
32necon2abid 2823 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝐵𝐴)) → ((𝑥𝑅𝐵𝐵𝑅𝑥) ↔ 𝑥𝐵))
43anass1rs 844 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) ↔ 𝑥𝐵))
5 breldmg 5238 . . . . . . . . . 10 ((𝑥𝐴𝐵𝐴𝑥𝑅𝐵) → 𝑥 ∈ dom 𝑅)
653expia 1258 . . . . . . . . 9 ((𝑥𝐴𝐵𝐴) → (𝑥𝑅𝐵𝑥 ∈ dom 𝑅))
76ancoms 467 . . . . . . . 8 ((𝐵𝐴𝑥𝐴) → (𝑥𝑅𝐵𝑥 ∈ dom 𝑅))
8 brelrng 5262 . . . . . . . . 9 ((𝐵𝐴𝑥𝐴𝐵𝑅𝑥) → 𝑥 ∈ ran 𝑅)
983expia 1258 . . . . . . . 8 ((𝐵𝐴𝑥𝐴) → (𝐵𝑅𝑥𝑥 ∈ ran 𝑅))
107, 9orim12d 878 . . . . . . 7 ((𝐵𝐴𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → (𝑥 ∈ dom 𝑅𝑥 ∈ ran 𝑅)))
11 elun 3714 . . . . . . 7 (𝑥 ∈ (dom 𝑅 ∪ ran 𝑅) ↔ (𝑥 ∈ dom 𝑅𝑥 ∈ ran 𝑅))
1210, 11syl6ibr 240 . . . . . 6 ((𝐵𝐴𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1312adantll 745 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
144, 13sylbird 248 . . . 4 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1514expimpd 626 . . 3 ((𝑅 Or 𝐴𝐵𝐴) → ((𝑥𝐴𝑥𝐵) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
161, 15syl5bi 230 . 2 ((𝑅 Or 𝐴𝐵𝐴) → (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1716ssrdv 3573 1 ((𝑅 Or 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382  wcel 1976  wne 2779  cdif 3536  cun 3537  wss 3539  {csn 4124   class class class wbr 4577   Or wor 4947  dom cdm 5027  ran crn 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-po 4948  df-so 4949  df-cnv 5035  df-dm 5037  df-rn 5038
This theorem is referenced by:  sofld  5485  soex  6979
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