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Theorem poslubmo 17067
Description: Least upper bounds in a poset are unique if they exist. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
poslubmo.l = (le‘𝐾)
poslubmo.b 𝐵 = (Base‘𝐾)
Assertion
Ref Expression
poslubmo ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))
Distinct variable groups:   𝑥, ,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem poslubmo
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simplrr 800 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑤𝐵)
2 simprlr 802 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧))
3 simprrl 803 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ∀𝑦𝑆 𝑦 𝑤)
4 breq2 4617 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑦 𝑧𝑦 𝑤))
54ralbidv 2980 . . . . . . . 8 (𝑧 = 𝑤 → (∀𝑦𝑆 𝑦 𝑧 ↔ ∀𝑦𝑆 𝑦 𝑤))
6 breq2 4617 . . . . . . . 8 (𝑧 = 𝑤 → (𝑥 𝑧𝑥 𝑤))
75, 6imbi12d 334 . . . . . . 7 (𝑧 = 𝑤 → ((∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) ↔ (∀𝑦𝑆 𝑦 𝑤𝑥 𝑤)))
87rspcv 3291 . . . . . 6 (𝑤𝐵 → (∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) → (∀𝑦𝑆 𝑦 𝑤𝑥 𝑤)))
91, 2, 3, 8syl3c 66 . . . . 5 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑥 𝑤)
10 simplrl 799 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑥𝐵)
11 simprrr 804 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))
12 simprll 801 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ∀𝑦𝑆 𝑦 𝑥)
13 breq2 4617 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑦 𝑧𝑦 𝑥))
1413ralbidv 2980 . . . . . . . 8 (𝑧 = 𝑥 → (∀𝑦𝑆 𝑦 𝑧 ↔ ∀𝑦𝑆 𝑦 𝑥))
15 breq2 4617 . . . . . . . 8 (𝑧 = 𝑥 → (𝑤 𝑧𝑤 𝑥))
1614, 15imbi12d 334 . . . . . . 7 (𝑧 = 𝑥 → ((∀𝑦𝑆 𝑦 𝑧𝑤 𝑧) ↔ (∀𝑦𝑆 𝑦 𝑥𝑤 𝑥)))
1716rspcv 3291 . . . . . 6 (𝑥𝐵 → (∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧) → (∀𝑦𝑆 𝑦 𝑥𝑤 𝑥)))
1810, 11, 12, 17syl3c 66 . . . . 5 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑤 𝑥)
19 poslubmo.b . . . . . . . 8 𝐵 = (Base‘𝐾)
20 poslubmo.l . . . . . . . 8 = (le‘𝐾)
2119, 20posasymb 16873 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑤𝐵) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
22213expb 1263 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑤𝐵)) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
2322ad4ant13 1289 . . . . 5 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
249, 18, 23mpbi2and 955 . . . 4 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑥 = 𝑤)
2524ex 450 . . 3 (((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) → (((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))) → 𝑥 = 𝑤))
2625ralrimivva 2965 . 2 ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∀𝑥𝐵𝑤𝐵 (((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))) → 𝑥 = 𝑤))
27 breq2 4617 . . . . 5 (𝑥 = 𝑤 → (𝑦 𝑥𝑦 𝑤))
2827ralbidv 2980 . . . 4 (𝑥 = 𝑤 → (∀𝑦𝑆 𝑦 𝑥 ↔ ∀𝑦𝑆 𝑦 𝑤))
29 breq1 4616 . . . . . 6 (𝑥 = 𝑤 → (𝑥 𝑧𝑤 𝑧))
3029imbi2d 330 . . . . 5 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) ↔ (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))
3130ralbidv 2980 . . . 4 (𝑥 = 𝑤 → (∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) ↔ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))
3228, 31anbi12d 746 . . 3 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ↔ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))))
3332rmo4 3381 . 2 (∃*𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ↔ ∀𝑥𝐵𝑤𝐵 (((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))) → 𝑥 = 𝑤))
3426, 33sylibr 224 1 ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  ∃*wrmo 2910  wss 3555   class class class wbr 4613  cfv 5847  Basecbs 15781  lecple 15869  Posetcpo 16861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-preset 16849  df-poset 16867
This theorem is referenced by:  poslubd  17069
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