Theorem somo 5598

Description: A totally ordered set has at most one minimal element. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
somo	⊢ (𝑅 Or 𝐴 → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem somo

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5106	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑥𝑅𝑧))
2	1	notbid 321	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑥𝑅𝑧))
3	2	rspcv 3573	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧 → ¬ 𝑥𝑅𝑧))
4		breq1 5106	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
5	4	notbid 321	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑧𝑅𝑥))
6	5	rspcv 3573	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 → ¬ 𝑧𝑅𝑥))
7	3, 6	im2anan9 632	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
8	7	ancomsd 471	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
9	8	imp 412	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧)) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
10		ioran 999	. . . . . . 7 ⊢ (¬ (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑥) ↔ (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
11		solin 5586	. . . . . . . . . 10 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
12		df-3or 1104	. . . . . . . . . 10 ⊢ ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
13	11, 12	sylib 221	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
14		or32 939	. . . . . . . . 9 ⊢ (((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧) ∨ 𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧 ∨ 𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
15	13, 14	sylib 221	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑧 ∨ 𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
16	15	ord 878	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (¬ (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑥) → 𝑥 = 𝑧))
17	10, 16	biimtrrid 246	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥) → 𝑥 = 𝑧))
18	9, 17	syl5 35	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧)) → 𝑥 = 𝑧))
19	18	exp4b 436	. . . 4 ⊢ (𝑅 Or 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))))
20	19	pm2.43d 54	. . 3 ⊢ (𝑅 Or 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧)))
21	20	ralrimivv 3204	. 2 ⊢ (𝑅 Or 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
22		breq2 5107	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
23	22	notbid 321	. . . 4 ⊢ (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
24	23	ralbidv 3186	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧))
25	24	rmo4 3688	. 2 ⊢ (∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
26	21, 25	sylibr 237	1 ⊢ (𝑅 Or 𝐴 → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∈ wcel 2145  ∀wral 3077  ∃*wrmo 3365   class class class wbr 5103   Or wor 5558

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 2147  ax-9 2155  ax-ext 2733

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-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rmo 3366  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-so 5560

This theorem is used by:  wereu  5647  wereu2  5648  nomaxmo  28037  nominmo  28038