Theorem sossfld 6133

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 4738	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2		sotrieq 5555	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → (𝑥 = 𝐵 ↔ ¬ (𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥)))
3	2	necon2abid 2970	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) ↔ 𝑥 ≠ 𝐵))
4	3	anass1rs 655	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) ↔ 𝑥 ≠ 𝐵))
5		breldmg 5849	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝑥𝑅𝐵) → 𝑥 ∈ dom 𝑅)
6	5	3expia 1121	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥𝑅𝐵 → 𝑥 ∈ dom 𝑅))
7	6	ancoms 458	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝐵 → 𝑥 ∈ dom 𝑅))
8		brelrng 5881	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵𝑅𝑥) → 𝑥 ∈ ran 𝑅)
9	8	3expia 1121	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝑥 → 𝑥 ∈ ran 𝑅))
10	7, 9	orim12d 966	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) → (𝑥 ∈ dom 𝑅 ∨ 𝑥 ∈ ran 𝑅)))
11		elun 4103	. . . . . . 7 ⊢ (𝑥 ∈ (dom 𝑅 ∪ ran 𝑅) ↔ (𝑥 ∈ dom 𝑅 ∨ 𝑥 ∈ ran 𝑅))
12	10, 11	imbitrrdi 252	. . . . . 6 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
13	12	adantll 714	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
14	4, 13	sylbird 260	. . . 4 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≠ 𝐵 → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
15	14	expimpd 453	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
16	1, 15	biimtrid 242	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
17	16	ssrdv 3940	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∈ wcel 2111   ≠ wne 2928   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  {csn 4576   class class class wbr 5091   Or wor 5523  dom cdm 5616  ran crn 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-po 5524  df-so 5525  df-cnv 5624  df-dm 5626  df-rn 5627

This theorem is referenced by:  sofld  6134  soex  7851