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Theorem somo 4993
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 4586 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝑅𝑧𝑥𝑅𝑧))
21notbid 307 . . . . . . . . . 10 (𝑦 = 𝑥 → (¬ 𝑦𝑅𝑧 ↔ ¬ 𝑥𝑅𝑧))
32rspcv 3278 . . . . . . . . 9 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑧 → ¬ 𝑥𝑅𝑧))
4 breq1 4586 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
54notbid 307 . . . . . . . . . 10 (𝑦 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑧𝑅𝑥))
65rspcv 3278 . . . . . . . . 9 (𝑧𝐴 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 → ¬ 𝑧𝑅𝑥))
73, 6im2anan9 876 . . . . . . . 8 ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑧 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
87ancomsd 469 . . . . . . 7 ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥)))
98imp 444 . . . . . 6 (((𝑥𝐴𝑧𝐴) ∧ (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧)) → (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
10 ioran 510 . . . . . . 7 (¬ (𝑥𝑅𝑧𝑧𝑅𝑥) ↔ (¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥))
11 solin 4982 . . . . . . . . . 10 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥))
12 df-3or 1032 . . . . . . . . . 10 ((𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
1311, 12sylib 207 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥))
14 or32 548 . . . . . . . . 9 (((𝑥𝑅𝑧𝑥 = 𝑧) ∨ 𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑧𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
1513, 14sylib 207 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → ((𝑥𝑅𝑧𝑧𝑅𝑥) ∨ 𝑥 = 𝑧))
1615ord 391 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → (¬ (𝑥𝑅𝑧𝑧𝑅𝑥) → 𝑥 = 𝑧))
1710, 16syl5bir 232 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → ((¬ 𝑥𝑅𝑧 ∧ ¬ 𝑧𝑅𝑥) → 𝑥 = 𝑧))
189, 17syl5 33 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝐴)) → (((𝑥𝐴𝑧𝐴) ∧ (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧)) → 𝑥 = 𝑧))
1918exp4b 630 . . . 4 (𝑅 Or 𝐴 → ((𝑥𝐴𝑧𝐴) → ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))))
2019pm2.43d 51 . . 3 (𝑅 Or 𝐴 → ((𝑥𝐴𝑧𝐴) → ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧)))
2120ralrimivv 2953 . 2 (𝑅 Or 𝐴 → ∀𝑥𝐴𝑧𝐴 ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
22 breq2 4587 . . . . 5 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
2322notbid 307 . . . 4 (𝑥 = 𝑧 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑧))
2423ralbidv 2969 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧))
2524rmo4 3366 . 2 (∃*𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑥𝐴𝑧𝐴 ((∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑧) → 𝑥 = 𝑧))
2621, 25sylibr 223 1 (𝑅 Or 𝐴 → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3o 1030  wcel 1977  wral 2896  ∃*wrmo 2899   class class class wbr 4583   Or wor 4958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rmo 2904  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-so 4960
This theorem is referenced by:  wereu  5034  wereu2  5035
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