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Theorem somo 5583
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 5109 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝑅𝑧𝑥𝑅𝑧))
21notbid 318 . . . . . . . . . 10 (𝑦 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑥𝑅𝑧))
32rspcv 3578 . . . . . . . . 9 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑧 → ¬ 𝑥𝑅𝑧))
4 breq1 5109 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
54notbid 318 . . . . . . . . . 10 (𝑦 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑧𝑅𝑥))
65rspcv 3578 . . . . . . . . 9 (𝑧𝐴 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 → ¬ 𝑧𝑅𝑥))
73, 6im2anan9 621 . . . . . . . 8 ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑧 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
87ancomsd 467 . . . . . . 7 ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
98imp 408 . . . . . 6 (((𝑥𝐴𝑧𝐴) ∧ (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧)) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
10 ioran 983 . . . . . . 7 (¬ (𝑥𝑅𝑧𝑧𝑅𝑥) ↔ (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
11 solin 5571 . . . . . . . . . 10 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
12 df-3or 1089 . . . . . . . . . 10 ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
1311, 12sylib 217 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
14 or32 925 . . . . . . . . 9 (((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
1513, 14sylib 217 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
1615ord 863 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → (¬ (𝑥𝑅𝑧𝑧𝑅𝑥) → 𝑥 = 𝑧))
1710, 16biimtrrid 242 . . . . . 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 432 . . . 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 3196 . 2 (𝑅 Or 𝐴 → ∀𝑥𝐴𝑧𝐴 ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
22 breq2 5110 . . . . 5 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
2322notbid 318 . . . 4 (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
2423ralbidv 3175 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧))
2524rmo4 3689 . 2 (∃*𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑥𝐴𝑧𝐴 ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
2621, 25sylibr 233 1 (𝑅 Or 𝐴 → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846  w3o 1087  wcel 2107  wral 3065  ∃*wrmo 3353   class class class wbr 5106   Or wor 5545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rmo 3354  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-so 5547
This theorem is referenced by:  wereu  5630  wereu2  5631  nomaxmo  27049  nominmo  27050
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