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Theorem sossfld 6036
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 4711 . . 3 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
2 sotrieq 5495 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝐵𝐴)) → (𝑥 = 𝐵 ↔ ¬ (𝑥𝑅𝐵𝐵𝑅𝑥)))
32necon2abid 3055 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝐵𝐴)) → ((𝑥𝑅𝐵𝐵𝑅𝑥) ↔ 𝑥𝐵))
43anass1rs 651 . . . . 5 (((𝑅 Or 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) ↔ 𝑥𝐵))
5 breldmg 5771 . . . . . . . . . 10 ((𝑥𝐴𝐵𝐴𝑥𝑅𝐵) → 𝑥 ∈ dom 𝑅)
653expia 1113 . . . . . . . . 9 ((𝑥𝐴𝐵𝐴) → (𝑥𝑅𝐵𝑥 ∈ dom 𝑅))
76ancoms 459 . . . . . . . 8 ((𝐵𝐴𝑥𝐴) → (𝑥𝑅𝐵𝑥 ∈ dom 𝑅))
8 brelrng 5804 . . . . . . . . 9 ((𝐵𝐴𝑥𝐴𝐵𝑅𝑥) → 𝑥 ∈ ran 𝑅)
983expia 1113 . . . . . . . 8 ((𝐵𝐴𝑥𝐴) → (𝐵𝑅𝑥𝑥 ∈ ran 𝑅))
107, 9orim12d 958 . . . . . . 7 ((𝐵𝐴𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → (𝑥 ∈ dom 𝑅𝑥 ∈ ran 𝑅)))
11 elun 4122 . . . . . . 7 (𝑥 ∈ (dom 𝑅 ∪ ran 𝑅) ↔ (𝑥 ∈ dom 𝑅𝑥 ∈ ran 𝑅))
1210, 11syl6ibr 253 . . . . . 6 ((𝐵𝐴𝑥𝐴) → ((𝑥𝑅𝐵𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1312adantll 710 . . . . 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 454 . . 3 ((𝑅 Or 𝐴𝐵𝐴) → ((𝑥𝐴𝑥𝐵) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
161, 15syl5bi 243 . 2 ((𝑅 Or 𝐴𝐵𝐴) → (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
1716ssrdv 3970 1 ((𝑅 Or 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  wcel 2105  wne 3013  cdif 3930  cun 3931  wss 3933  {csn 4557   class class class wbr 5057   Or wor 5466  dom cdm 5548  ran crn 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-po 5467  df-so 5468  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by:  sofld  6037  soex  7615
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