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Theorem somo 5509
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 5066 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝑅𝑧𝑥𝑅𝑧))
21notbid 319 . . . . . . . . . 10 (𝑦 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑥𝑅𝑧))
32rspcv 3622 . . . . . . . . 9 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑧 → ¬ 𝑥𝑅𝑧))
4 breq1 5066 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
54notbid 319 . . . . . . . . . 10 (𝑦 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑧𝑅𝑥))
65rspcv 3622 . . . . . . . . 9 (𝑧𝐴 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 → ¬ 𝑧𝑅𝑥))
73, 6im2anan9 619 . . . . . . . 8 ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑧 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
87ancomsd 466 . . . . . . 7 ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
98imp 407 . . . . . 6 (((𝑥𝐴𝑧𝐴) ∧ (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧)) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
10 ioran 979 . . . . . . 7 (¬ (𝑥𝑅𝑧𝑧𝑅𝑥) ↔ (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
11 solin 5497 . . . . . . . . . 10 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
12 df-3or 1082 . . . . . . . . . 10 ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
1311, 12sylib 219 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
14 or32 921 . . . . . . . . 9 (((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
1513, 14sylib 219 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
1615ord 860 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → (¬ (𝑥𝑅𝑧𝑧𝑅𝑥) → 𝑥 = 𝑧))
1710, 16syl5bir 244 . . . . . 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 431 . . . 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 3195 . 2 (𝑅 Or 𝐴 → ∀𝑥𝐴𝑧𝐴 ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
22 breq2 5067 . . . . 5 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
2322notbid 319 . . . 4 (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
2423ralbidv 3202 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧))
2524rmo4 3725 . 2 (∃*𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑥𝐴𝑧𝐴 ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
2621, 25sylibr 235 1 (𝑅 Or 𝐴 → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 843  w3o 1080  wcel 2107  wral 3143  ∃*wrmo 3146   class class class wbr 5063   Or wor 5472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rmo 3151  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-so 5474
This theorem is referenced by:  wereu  5550  wereu2  5551  nomaxmo  33085
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