Theorem sossfld 6175

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 4782	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2		sotrieq 5607	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → (𝑥 = 𝐵 ↔ ¬ (𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥)))
3	2	necon2abid 2975	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) ↔ 𝑥 ≠ 𝐵))
4	3	anass1rs 652	. . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) ↔ 𝑥 ≠ 𝐵))
5		breldmg 5899	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝑥𝑅𝐵) → 𝑥 ∈ dom 𝑅)
6	5	3expia 1118	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥𝑅𝐵 → 𝑥 ∈ dom 𝑅))
7	6	ancoms 458	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝐵 → 𝑥 ∈ dom 𝑅))
8		brelrng 5930	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵𝑅𝑥) → 𝑥 ∈ ran 𝑅)
9	8	3expia 1118	. . . . . . . 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 4140	. . . . . . 7 ⊢ (𝑥 ∈ (dom 𝑅 ∪ ran 𝑅) ↔ (𝑥 ∈ dom 𝑅 ∨ 𝑥 ∈ ran 𝑅))
12	10, 11	imbitrrdi 251	. . . . . 6 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥𝑅𝐵 ∨ 𝐵𝑅𝑥) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
13	12	adantll 711	. . . . 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 241	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅)))
17	16	ssrdv 3980	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∈ wcel 2098   ≠ wne 2932   ∖ cdif 3937   ∪ cun 3938   ⊆ wss 3940  {csn 4620   class class class wbr 5138   Or wor 5577  dom cdm 5666  ran crn 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-po 5578  df-so 5579  df-cnv 5674  df-dm 5676  df-rn 5677

This theorem is referenced by:  sofld  6176  soex  7905