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Theorem somo 5483
 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 → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem somo
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 breq1 5039 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝑅𝑧𝑥𝑅𝑧))
21notbid 321 . . . . . . . . . 10 (𝑦 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑥𝑅𝑧))
32rspcv 3538 . . . . . . . . 9 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑧 → ¬ 𝑥𝑅𝑧))
4 breq1 5039 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
54notbid 321 . . . . . . . . . 10 (𝑦 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑧𝑅𝑥))
65rspcv 3538 . . . . . . . . 9 (𝑧𝐴 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 → ¬ 𝑧𝑅𝑥))
73, 6im2anan9 622 . . . . . . . 8 ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑧 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
87ancomsd 469 . . . . . . 7 ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
98imp 410 . . . . . 6 (((𝑥𝐴𝑧𝐴) ∧ (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧)) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
10 ioran 981 . . . . . . 7 (¬ (𝑥𝑅𝑧𝑧𝑅𝑥) ↔ (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
11 solin 5471 . . . . . . . . . 10 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
12 df-3or 1085 . . . . . . . . . 10 ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
1311, 12sylib 221 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
14 or32 923 . . . . . . . . 9 (((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
1513, 14sylib 221 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
1615ord 861 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → (¬ (𝑥𝑅𝑧𝑧𝑅𝑥) → 𝑥 = 𝑧))
1710, 16syl5bir 246 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → ((¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥) → 𝑥 = 𝑧))
189, 17syl5 34 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → (((𝑥𝐴𝑧𝐴) ∧ (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧)) → 𝑥 = 𝑧))
1918exp4b 434 . . . 4 (𝑅 Or 𝐴 → ((𝑥𝐴𝑧𝐴) → ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))))
2019pm2.43d 53 . . 3 (𝑅 Or 𝐴 → ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧)))
2120ralrimivv 3119 . 2 (𝑅 Or 𝐴 → ∀𝑥𝐴𝑧𝐴 ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
22 breq2 5040 . . . . 5 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
2322notbid 321 . . . 4 (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
2423ralbidv 3126 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧))
2524rmo4 3646 . 2 (∃*𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑥𝐴𝑧𝐴 ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
2621, 25sylibr 237 1 (𝑅 Or 𝐴 → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ∈ wcel 2111  ∀wral 3070  ∃*wrmo 3073   class class class wbr 5036   Or wor 5446 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 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-mo 2557  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rmo 3078  df-v 3411  df-un 3865  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-so 5448 This theorem is referenced by:  wereu  5524  wereu2  5525  nomaxmo  33498  nominmo  33499
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