Theorem somo 5531

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 5073	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑥𝑅𝑧))
2	1	notbid 317	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑥𝑅𝑧))
3	2	rspcv 3547	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧 → ¬ 𝑥𝑅𝑧))
4		breq1 5073	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
5	4	notbid 317	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑧𝑅𝑥))
6	5	rspcv 3547	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 → ¬ 𝑧𝑅𝑥))
7	3, 6	im2anan9 619	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
8	7	ancomsd 465	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
9	8	imp 406	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧)) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
10		ioran 980	. . . . . . 7 ⊢ (¬ (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑥) ↔ (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
11		solin 5519	. . . . . . . . . 10 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
12		df-3or 1086	. . . . . . . . . 10 ⊢ ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
13	11, 12	sylib 217	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
14		or32 922	. . . . . . . . 9 ⊢ (((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧) ∨ 𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧 ∨ 𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
15	13, 14	sylib 217	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑧 ∨ 𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
16	15	ord 860	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (¬ (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑥) → 𝑥 = 𝑧))
17	10, 16	syl5bir 242	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥) → 𝑥 = 𝑧))
18	9, 17	syl5 34	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧)) → 𝑥 = 𝑧))
19	18	exp4b 430	. . . 4 ⊢ (𝑅 Or 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))))
20	19	pm2.43d 53	. . 3 ⊢ (𝑅 Or 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧)))
21	20	ralrimivv 3113	. 2 ⊢ (𝑅 Or 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
22		breq2 5074	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
23	22	notbid 317	. . . 4 ⊢ (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
24	23	ralbidv 3120	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧))
25	24	rmo4 3660	. 2 ⊢ (∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
26	21, 25	sylibr 233	1 ⊢ (𝑅 Or 𝐴 → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   ∨ w3o 1084   ∈ wcel 2108  ∀wral 3063  ∃*wrmo 3066   class class class wbr 5070   Or wor 5493

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

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-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rmo 3071  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-so 5495

This theorem is referenced by:  wereu  5576  wereu2  5577  nomaxmo  33828  nominmo  33829