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Theorem poslubmo 17354
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 simprrl 766 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ∀𝑦𝑆 𝑦 𝑤)
2 breq2 4791 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑦 𝑧𝑦 𝑤))
32ralbidv 3135 . . . . . . . 8 (𝑧 = 𝑤 → (∀𝑦𝑆 𝑦 𝑧 ↔ ∀𝑦𝑆 𝑦 𝑤))
4 breq2 4791 . . . . . . . 8 (𝑧 = 𝑤 → (𝑥 𝑧𝑥 𝑤))
53, 4imbi12d 333 . . . . . . 7 (𝑧 = 𝑤 → ((∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) ↔ (∀𝑦𝑆 𝑦 𝑤𝑥 𝑤)))
6 simprlr 765 . . . . . . 7 ((((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))
7 simplrr 763 . . . . . . 7 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑤𝐵)
85, 6, 7rspcdva 3466 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → (∀𝑦𝑆 𝑦 𝑤𝑥 𝑤))
91, 8mpd 15 . . . . 5 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑥 𝑤)
10 simprll 764 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ∀𝑦𝑆 𝑦 𝑥)
11 breq2 4791 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑦 𝑧𝑦 𝑥))
1211ralbidv 3135 . . . . . . . 8 (𝑧 = 𝑥 → (∀𝑦𝑆 𝑦 𝑧 ↔ ∀𝑦𝑆 𝑦 𝑥))
13 breq2 4791 . . . . . . . 8 (𝑧 = 𝑥 → (𝑤 𝑧𝑤 𝑥))
1412, 13imbi12d 333 . . . . . . 7 (𝑧 = 𝑥 → ((∀𝑦𝑆 𝑦 𝑧𝑤 𝑧) ↔ (∀𝑦𝑆 𝑦 𝑥𝑤 𝑥)))
15 simprrr 767 . . . . . . 7 ((((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))
16 simplrl 762 . . . . . . 7 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑥𝐵)
1714, 15, 16rspcdva 3466 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → (∀𝑦𝑆 𝑦 𝑥𝑤 𝑥))
1810, 17mpd 15 . . . . 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 17160 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑤𝐵) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
22213expb 1113 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑤𝐵)) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
2322ad4ant13 1206 . . . . 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 691 . . . 4 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑥 = 𝑤)
2524ex 397 . . 3 (((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) → (((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))) → 𝑥 = 𝑤))
2625ralrimivva 3120 . 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 4791 . . . . 5 (𝑥 = 𝑤 → (𝑦 𝑥𝑦 𝑤))
2827ralbidv 3135 . . . 4 (𝑥 = 𝑤 → (∀𝑦𝑆 𝑦 𝑥 ↔ ∀𝑦𝑆 𝑦 𝑤))
29 breq1 4790 . . . . . 6 (𝑥 = 𝑤 → (𝑥 𝑧𝑤 𝑧))
3029imbi2d 329 . . . . 5 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) ↔ (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))
3130ralbidv 3135 . . . 4 (𝑥 = 𝑤 → (∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) ↔ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))
3228, 31anbi12d 616 . . 3 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ↔ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))))
3332rmo4 3551 . 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 382   = wceq 1631  wcel 2145  wral 3061  ∃*wrmo 3064  wss 3723   class class class wbr 4787  cfv 6030  Basecbs 16064  lecple 16156  Posetcpo 17148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-preset 17136  df-poset 17154
This theorem is referenced by:  poslubd  17356
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