Theorem sossfld 6189

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 4758	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2		sotrieq 5605	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → (𝑥 = 𝐵 ↔ ¬ (𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥)))
3	2	necon2abid 3003	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) ↔ 𝑥 ≠ 𝐵))
4	3	anass1rs 668	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) ↔ 𝑥 ≠ 𝐵))
5		breldmg 5904	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝑥𝑅𝐵) → 𝑥 ∈ dom 𝑅)
6	5	3expia 1139	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥𝑅𝐵 → 𝑥 ∈ dom 𝑅))
7	6	ancoms 464	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝐵 → 𝑥 ∈ dom 𝑅))
8		brelrng 5936	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵𝑅𝑥) → 𝑥 ∈ ran 𝑅)
9	8	3expia 1139	. . . . . . . 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 4110	. . . . . . 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 727	. . . . 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 459	. . 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 3946	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∈ wcel 2146   ≠ wne 2961   ∖ cdif 3905   ∪ cun 3906   ⊆ wss 3908  {csn 4594   class class class wbr 5114   Or wor 5573  dom cdm 5666  ran crn 5667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-po 5574  df-so 5575  df-cnv 5674  df-dm 5676  df-rn 5677

This theorem is used by:  sofld  6190  soex  7927