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Theorem sossfld 6020
 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 4680 . . 3 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
2 sotrieq 5475 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝐵𝐴)) → (𝑥 = 𝐵 ↔ ¬ (𝑥𝑅𝐵𝐵𝑅𝑥)))
32necon2abid 2993 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝐵𝐴)) → ((𝑥𝑅𝐵𝐵𝑅𝑥) ↔ 𝑥𝐵))
43anass1rs 654 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) ↔ 𝑥𝐵))
5 breldmg 5755 . . . . . . . . . 10 ((𝑥𝐴𝐵𝐴𝑥𝑅𝐵) → 𝑥 ∈ dom 𝑅)
653expia 1118 . . . . . . . . 9 ((𝑥𝐴𝐵𝐴) → (𝑥𝑅𝐵𝑥 ∈ dom 𝑅))
76ancoms 462 . . . . . . . 8 ((𝐵𝐴𝑥𝐴) → (𝑥𝑅𝐵𝑥 ∈ dom 𝑅))
8 brelrng 5787 . . . . . . . . 9 ((𝐵𝐴𝑥𝐴𝐵𝑅𝑥) → 𝑥 ∈ ran 𝑅)
983expia 1118 . . . . . . . 8 ((𝐵𝐴𝑥𝐴) → (𝐵𝑅𝑥𝑥 ∈ ran 𝑅))
107, 9orim12d 962 . . . . . . 7 ((𝐵𝐴𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → (𝑥 ∈ dom 𝑅𝑥 ∈ ran 𝑅)))
11 elun 4056 . . . . . . 7 (𝑥 ∈ (dom 𝑅 ∪ ran 𝑅) ↔ (𝑥 ∈ dom 𝑅𝑥 ∈ ran 𝑅))
1210, 11syl6ibr 255 . . . . . 6 ((𝐵𝐴𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1312adantll 713 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
144, 13sylbird 263 . . . 4 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1514expimpd 457 . . 3 ((𝑅 Or 𝐴𝐵𝐴) → ((𝑥𝐴𝑥𝐵) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
161, 15syl5bi 245 . 2 ((𝑅 Or 𝐴𝐵𝐴) → (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1716ssrdv 3900 1 ((𝑅 Or 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∈ wcel 2111   ≠ wne 2951   ∖ cdif 3857   ∪ cun 3858   ⊆ wss 3860  {csn 4525   class class class wbr 5036   Or wor 5446  dom cdm 5528  ran crn 5529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-po 5447  df-so 5448  df-cnv 5536  df-dm 5538  df-rn 5539 This theorem is referenced by:  sofld  6021  soex  7637
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