Theorem sossfld 6176

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 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))

Proof of Theorem sossfld

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsn 4749	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2		sotrieq 5591	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → (𝑥 = 𝐵 ↔ ¬ (𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥)))
3	2	necon2abid 3002	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) ↔ 𝑥 ≠ 𝐵))
4	3	anass1rs 667	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) ↔ 𝑥 ≠ 𝐵))
5		breldmg 5890	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝑥𝑅𝐵) → 𝑥 ∈ dom 𝑅)
6	5	3expia 1137	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥𝑅𝐵 → 𝑥 ∈ dom 𝑅))
7	6	ancoms 463	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝐵 → 𝑥 ∈ dom 𝑅))
8		brelrng 5922	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵𝑅𝑥) → 𝑥 ∈ ran 𝑅)
9	8	3expia 1137	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝑥 → 𝑥 ∈ ran 𝑅))
10	7, 9	orim12d 979	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) → (𝑥 ∈ dom 𝑅 ∨ 𝑥 ∈ ran 𝑅)))
11		elun 4109	. . . . . . 7 ⊢ (𝑥 ∈ (dom 𝑅 ∪ ran 𝑅) ↔ (𝑥 ∈ dom 𝑅 ∨ 𝑥 ∈ ran 𝑅))
12	10, 11	imbitrrdi 255	. . . . . 6 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
13	12	adantll 726	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
14	4, 13	sylbird 263	. . . 4 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≠ 𝐵 → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
15	14	expimpd 458	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
16	1, 15	biimtrid 245	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
17	16	ssrdv 3945	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∈ wcel 2145   ≠ wne 2960   ∖ cdif 3904   ∪ cun 3905   ⊆ wss 3907  {csn 4585   class class class wbr 5105   Or wor 5559  dom cdm 5652  ran crn 5653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-po 5560  df-so 5561  df-cnv 5660  df-dm 5662  df-rn 5663

This theorem is referenced by:  sofld  6177  soex  7906