Theorem somo 5474

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 5033	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑥𝑅𝑧))
2	1	notbid 321	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑥𝑅𝑧))
3	2	rspcv 3566	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧 → ¬ 𝑥𝑅𝑧))
4		breq1 5033	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
5	4	notbid 321	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑧𝑅𝑥))
6	5	rspcv 3566	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 → ¬ 𝑧𝑅𝑥))
7	3, 6	im2anan9 622	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
8	7	ancomsd 469	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
9	8	imp 410	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧)) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
10		ioran 981	. . . . . . 7 ⊢ (¬ (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑥) ↔ (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
11		solin 5462	. . . . . . . . . 10 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
12		df-3or 1085	. . . . . . . . . 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 923	. . . . . . . . 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 861	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (¬ (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑥) → 𝑥 = 𝑧))
17	10, 16	syl5bir 246	. . . . . 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 434	. . . 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 3155	. 2 ⊢ (𝑅 Or 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
22		breq2 5034	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
23	22	notbid 321	. . . 4 ⊢ (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
24	23	ralbidv 3162	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧))
25	24	rmo4 3669	. 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)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ∈ wcel 2111  ∀wral 3106  ∃*wrmo 3109   class class class wbr 5030   Or wor 5437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-ex 1782  df-sb 2070  df-mo 2598  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rmo 3114  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-so 5439

This theorem is referenced by:  wereu  5515  wereu2  5516  nomaxmo  33314