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Theorem supmo 8703
 Description: Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by NM, 5-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
supmo.1 (𝜑𝑅 Or 𝐴)
Assertion
Ref Expression
supmo (𝜑 → ∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝑅,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem supmo
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 supmo.1 . . 3 (𝜑𝑅 Or 𝐴)
2 ancom 453 . . . . . . . 8 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦) ↔ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦))
32anbi2ci 615 . . . . . . 7 (((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦) ∧ (∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) ↔ ((∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦)))
4 an42 644 . . . . . . 7 (((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧))) ↔ ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦) ∧ (∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
5 an42 644 . . . . . . 7 (((∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦) ∧ (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦)) ↔ ((∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦)))
63, 4, 53bitr4i 295 . . . . . 6 (((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧))) ↔ ((∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦) ∧ (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦)))
7 ralnex 3177 . . . . . . . . . 10 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ¬ ∃𝑦𝐵 𝑥𝑅𝑦)
8 breq1 4926 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑦𝑅𝑤𝑥𝑅𝑤))
9 breq1 4926 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦𝑅𝑧𝑥𝑅𝑧))
109rexbidv 3236 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (∃𝑧𝐵 𝑦𝑅𝑧 ↔ ∃𝑧𝐵 𝑥𝑅𝑧))
118, 10imbi12d 337 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑤 → ∃𝑧𝐵 𝑥𝑅𝑧)))
1211rspcva 3527 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑥𝑅𝑤 → ∃𝑧𝐵 𝑥𝑅𝑧))
13 breq2 4927 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑥𝑅𝑦𝑥𝑅𝑧))
1413cbvrexv 3378 . . . . . . . . . . . 12 (∃𝑦𝐵 𝑥𝑅𝑦 ↔ ∃𝑧𝐵 𝑥𝑅𝑧)
1512, 14syl6ibr 244 . . . . . . . . . . 11 ((𝑥𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑥𝑅𝑤 → ∃𝑦𝐵 𝑥𝑅𝑦))
1615con3d 150 . . . . . . . . . 10 ((𝑥𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (¬ ∃𝑦𝐵 𝑥𝑅𝑦 → ¬ 𝑥𝑅𝑤))
177, 16syl5bi 234 . . . . . . . . 9 ((𝑥𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 → ¬ 𝑥𝑅𝑤))
1817expimpd 446 . . . . . . . 8 (𝑥𝐴 → ((∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦) → ¬ 𝑥𝑅𝑤))
1918ad2antrl 715 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → ((∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦) → ¬ 𝑥𝑅𝑤))
20 ralnex 3177 . . . . . . . . . 10 (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ↔ ¬ ∃𝑦𝐵 𝑤𝑅𝑦)
21 breq1 4926 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝑦𝑅𝑥𝑤𝑅𝑥))
22 breq1 4926 . . . . . . . . . . . . . . 15 (𝑦 = 𝑤 → (𝑦𝑅𝑧𝑤𝑅𝑧))
2322rexbidv 3236 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (∃𝑧𝐵 𝑦𝑅𝑧 ↔ ∃𝑧𝐵 𝑤𝑅𝑧))
2421, 23imbi12d 337 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)))
2524rspcva 3527 . . . . . . . . . . . 12 ((𝑤𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧))
26 breq2 4927 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑤𝑅𝑦𝑤𝑅𝑧))
2726cbvrexv 3378 . . . . . . . . . . . 12 (∃𝑦𝐵 𝑤𝑅𝑦 ↔ ∃𝑧𝐵 𝑤𝑅𝑧)
2825, 27syl6ibr 244 . . . . . . . . . . 11 ((𝑤𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑤𝑅𝑥 → ∃𝑦𝐵 𝑤𝑅𝑦))
2928con3d 150 . . . . . . . . . 10 ((𝑤𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (¬ ∃𝑦𝐵 𝑤𝑅𝑦 → ¬ 𝑤𝑅𝑥))
3020, 29syl5bi 234 . . . . . . . . 9 ((𝑤𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 → ¬ 𝑤𝑅𝑥))
3130expimpd 446 . . . . . . . 8 (𝑤𝐴 → ((∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦) → ¬ 𝑤𝑅𝑥))
3231ad2antll 716 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → ((∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦) → ¬ 𝑤𝑅𝑥))
3319, 32anim12d 599 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → (((∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦) ∧ (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦)) → (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
346, 33syl5bi 234 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → (((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧))) → (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
35 sotrieq2 5348 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → (𝑥 = 𝑤 ↔ (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
3634, 35sylibrd 251 . . . 4 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → (((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧))) → 𝑥 = 𝑤))
3736ralrimivva 3135 . . 3 (𝑅 Or 𝐴 → ∀𝑥𝐴𝑤𝐴 (((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧))) → 𝑥 = 𝑤))
381, 37syl 17 . 2 (𝜑 → ∀𝑥𝐴𝑤𝐴 (((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧))) → 𝑥 = 𝑤))
39 breq1 4926 . . . . . 6 (𝑥 = 𝑤 → (𝑥𝑅𝑦𝑤𝑅𝑦))
4039notbid 310 . . . . 5 (𝑥 = 𝑤 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
4140ralbidv 3141 . . . 4 (𝑥 = 𝑤 → (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦))
42 breq2 4927 . . . . . 6 (𝑥 = 𝑤 → (𝑦𝑅𝑥𝑦𝑅𝑤))
4342imbi1d 334 . . . . 5 (𝑥 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)))
4443ralbidv 3141 . . . 4 (𝑥 = 𝑤 → (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)))
4541, 44anbi12d 621 . . 3 (𝑥 = 𝑤 → ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧))))
4645rmo4 3629 . 2 (∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ ∀𝑥𝐴𝑤𝐴 (((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧))) → 𝑥 = 𝑤))
4738, 46sylibr 226 1 (𝜑 → ∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387   ∈ wcel 2048  ∀wral 3082  ∃wrex 3083  ∃*wrmo 3085   class class class wbr 4923   Or wor 5318 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rmo 3090  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-po 5319  df-so 5320 This theorem is referenced by:  supexd  8704  supeu  8705
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