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Theorem sossfld 6100
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 4726 . . 3 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
2 sotrieq 5539 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝐵𝐴)) → (𝑥 = 𝐵 ↔ ¬ (𝑥𝑅𝐵𝐵𝑅𝑥)))
32necon2abid 2984 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝐵𝐴)) → ((𝑥𝑅𝐵𝐵𝑅𝑥) ↔ 𝑥𝐵))
43anass1rs 653 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) ↔ 𝑥𝐵))
5 breldmg 5827 . . . . . . . . . 10 ((𝑥𝐴𝐵𝐴𝑥𝑅𝐵) → 𝑥 ∈ dom 𝑅)
653expia 1121 . . . . . . . . 9 ((𝑥𝐴𝐵𝐴) → (𝑥𝑅𝐵𝑥 ∈ dom 𝑅))
76ancoms 460 . . . . . . . 8 ((𝐵𝐴𝑥𝐴) → (𝑥𝑅𝐵𝑥 ∈ dom 𝑅))
8 brelrng 5858 . . . . . . . . 9 ((𝐵𝐴𝑥𝐴𝐵𝑅𝑥) → 𝑥 ∈ ran 𝑅)
983expia 1121 . . . . . . . 8 ((𝐵𝐴𝑥𝐴) → (𝐵𝑅𝑥𝑥 ∈ ran 𝑅))
107, 9orim12d 963 . . . . . . 7 ((𝐵𝐴𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → (𝑥 ∈ dom 𝑅𝑥 ∈ ran 𝑅)))
11 elun 4089 . . . . . . 7 (𝑥 ∈ (dom 𝑅 ∪ ran 𝑅) ↔ (𝑥 ∈ dom 𝑅𝑥 ∈ ran 𝑅))
1210, 11syl6ibr 253 . . . . . 6 ((𝐵𝐴𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1312adantll 712 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
144, 13sylbird 261 . . . 4 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1514expimpd 455 . . 3 ((𝑅 Or 𝐴𝐵𝐴) → ((𝑥𝐴𝑥𝐵) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
161, 15biimtrid 242 . 2 ((𝑅 Or 𝐴𝐵𝐴) → (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1716ssrdv 3932 1 ((𝑅 Or 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 845  wcel 2104  wne 2941  cdif 3889  cun 3890  wss 3892  {csn 4565   class class class wbr 5081   Or wor 5509  dom cdm 5596  ran crn 5597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-po 5510  df-so 5511  df-cnv 5604  df-dm 5606  df-rn 5607
This theorem is referenced by:  sofld  6101  soex  7796
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