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Theorem sossfld 5615
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 4350 . . 3 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
2 sotrieq 5091 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝐵𝐴)) → (𝑥 = 𝐵 ↔ ¬ (𝑥𝑅𝐵𝐵𝑅𝑥)))
32necon2abid 2865 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝐵𝐴)) → ((𝑥𝑅𝐵𝐵𝑅𝑥) ↔ 𝑥𝐵))
43anass1rs 866 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) ↔ 𝑥𝐵))
5 breldmg 5362 . . . . . . . . . 10 ((𝑥𝐴𝐵𝐴𝑥𝑅𝐵) → 𝑥 ∈ dom 𝑅)
653expia 1286 . . . . . . . . 9 ((𝑥𝐴𝐵𝐴) → (𝑥𝑅𝐵𝑥 ∈ dom 𝑅))
76ancoms 468 . . . . . . . 8 ((𝐵𝐴𝑥𝐴) → (𝑥𝑅𝐵𝑥 ∈ dom 𝑅))
8 brelrng 5387 . . . . . . . . 9 ((𝐵𝐴𝑥𝐴𝐵𝑅𝑥) → 𝑥 ∈ ran 𝑅)
983expia 1286 . . . . . . . 8 ((𝐵𝐴𝑥𝐴) → (𝐵𝑅𝑥𝑥 ∈ ran 𝑅))
107, 9orim12d 901 . . . . . . 7 ((𝐵𝐴𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → (𝑥 ∈ dom 𝑅𝑥 ∈ ran 𝑅)))
11 elun 3786 . . . . . . 7 (𝑥 ∈ (dom 𝑅 ∪ ran 𝑅) ↔ (𝑥 ∈ dom 𝑅𝑥 ∈ ran 𝑅))
1210, 11syl6ibr 242 . . . . . 6 ((𝐵𝐴𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1312adantll 750 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
144, 13sylbird 250 . . . 4 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1514expimpd 628 . . 3 ((𝑅 Or 𝐴𝐵𝐴) → ((𝑥𝐴𝑥𝐵) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
161, 15syl5bi 232 . 2 ((𝑅 Or 𝐴𝐵𝐴) → (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1716ssrdv 3642 1 ((𝑅 Or 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  wcel 2030  wne 2823  cdif 3604  cun 3605  wss 3607  {csn 4210   class class class wbr 4685   Or wor 5063  dom cdm 5143  ran crn 5144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  sofld  5616  soex  7151
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