Theorem sossfld 6078

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 4717	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2		sotrieq 5523	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → (𝑥 = 𝐵 ↔ ¬ (𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥)))
3	2	necon2abid 2985	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) ↔ 𝑥 ≠ 𝐵))
4	3	anass1rs 651	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) ↔ 𝑥 ≠ 𝐵))
5		breldmg 5807	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝑥𝑅𝐵) → 𝑥 ∈ dom 𝑅)
6	5	3expia 1119	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥𝑅𝐵 → 𝑥 ∈ dom 𝑅))
7	6	ancoms 458	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝐵 → 𝑥 ∈ dom 𝑅))
8		brelrng 5839	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵𝑅𝑥) → 𝑥 ∈ ran 𝑅)
9	8	3expia 1119	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝑥 → 𝑥 ∈ ran 𝑅))
10	7, 9	orim12d 961	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) → (𝑥 ∈ dom 𝑅 ∨ 𝑥 ∈ ran 𝑅)))
11		elun 4079	. . . . . . 7 ⊢ (𝑥 ∈ (dom 𝑅 ∪ ran 𝑅) ↔ (𝑥 ∈ dom 𝑅 ∨ 𝑥 ∈ ran 𝑅))
12	10, 11	syl6ibr 251	. . . . . 6 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
13	12	adantll 710	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
14	4, 13	sylbird 259	. . . 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	syl5bi 241	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
17	16	ssrdv 3923	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∈ wcel 2108   ≠ wne 2942   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  {csn 4558   class class class wbr 5070   Or wor 5493  dom cdm 5580  ran crn 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-po 5494  df-so 5495  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  sofld  6079  soex  7742