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Theorem supmo 8399
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 465 . . . . . . . 8 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦) ↔ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦))
32anbi2ci 732 . . . . . . 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 883 . . . . . . 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 883 . . . . . . 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 292 . . . . . 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 3021 . . . . . . . . . 10 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ¬ ∃𝑦𝐵 𝑥𝑅𝑦)
8 breq1 4688 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑦𝑅𝑤𝑥𝑅𝑤))
9 breq1 4688 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦𝑅𝑧𝑥𝑅𝑧))
109rexbidv 3081 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (∃𝑧𝐵 𝑦𝑅𝑧 ↔ ∃𝑧𝐵 𝑥𝑅𝑧))
118, 10imbi12d 333 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑤 → ∃𝑧𝐵 𝑥𝑅𝑧)))
1211rspcva 3338 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑥𝑅𝑤 → ∃𝑧𝐵 𝑥𝑅𝑧))
13 breq2 4689 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑥𝑅𝑦𝑥𝑅𝑧))
1413cbvrexv 3202 . . . . . . . . . . . 12 (∃𝑦𝐵 𝑥𝑅𝑦 ↔ ∃𝑧𝐵 𝑥𝑅𝑧)
1512, 14syl6ibr 242 . . . . . . . . . . 11 ((𝑥𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑥𝑅𝑤 → ∃𝑦𝐵 𝑥𝑅𝑦))
1615con3d 148 . . . . . . . . . 10 ((𝑥𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (¬ ∃𝑦𝐵 𝑥𝑅𝑦 → ¬ 𝑥𝑅𝑤))
177, 16syl5bi 232 . . . . . . . . 9 ((𝑥𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 → ¬ 𝑥𝑅𝑤))
1817expimpd 628 . . . . . . . 8 (𝑥𝐴 → ((∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦) → ¬ 𝑥𝑅𝑤))
1918ad2antrl 764 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → ((∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦) → ¬ 𝑥𝑅𝑤))
20 ralnex 3021 . . . . . . . . . 10 (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ↔ ¬ ∃𝑦𝐵 𝑤𝑅𝑦)
21 breq1 4688 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝑦𝑅𝑥𝑤𝑅𝑥))
22 breq1 4688 . . . . . . . . . . . . . . 15 (𝑦 = 𝑤 → (𝑦𝑅𝑧𝑤𝑅𝑧))
2322rexbidv 3081 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (∃𝑧𝐵 𝑦𝑅𝑧 ↔ ∃𝑧𝐵 𝑤𝑅𝑧))
2421, 23imbi12d 333 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)))
2524rspcva 3338 . . . . . . . . . . . 12 ((𝑤𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧))
26 breq2 4689 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑤𝑅𝑦𝑤𝑅𝑧))
2726cbvrexv 3202 . . . . . . . . . . . 12 (∃𝑦𝐵 𝑤𝑅𝑦 ↔ ∃𝑧𝐵 𝑤𝑅𝑧)
2825, 27syl6ibr 242 . . . . . . . . . . 11 ((𝑤𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑤𝑅𝑥 → ∃𝑦𝐵 𝑤𝑅𝑦))
2928con3d 148 . . . . . . . . . 10 ((𝑤𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (¬ ∃𝑦𝐵 𝑤𝑅𝑦 → ¬ 𝑤𝑅𝑥))
3020, 29syl5bi 232 . . . . . . . . 9 ((𝑤𝐴 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 → ¬ 𝑤𝑅𝑥))
3130expimpd 628 . . . . . . . 8 (𝑤𝐴 → ((∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦) → ¬ 𝑤𝑅𝑥))
3231ad2antll 765 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → ((∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦) → ¬ 𝑤𝑅𝑥))
3319, 32anim12d 585 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → (((∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦) ∧ (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ∧ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦)) → (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
346, 33syl5bi 232 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → (((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧))) → (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
35 sotrieq2 5092 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → (𝑥 = 𝑤 ↔ (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
3634, 35sylibrd 249 . . . 4 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑤𝐴)) → (((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧))) → 𝑥 = 𝑤))
3736ralrimivva 3000 . . 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 4688 . . . . . 6 (𝑥 = 𝑤 → (𝑥𝑅𝑦𝑤𝑅𝑦))
4039notbid 307 . . . . 5 (𝑥 = 𝑤 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
4140ralbidv 3015 . . . 4 (𝑥 = 𝑤 → (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦𝐵 ¬ 𝑤𝑅𝑦))
42 breq2 4689 . . . . . 6 (𝑥 = 𝑤 → (𝑦𝑅𝑥𝑦𝑅𝑤))
4342imbi1d 330 . . . . 5 (𝑥 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)))
4443ralbidv 3015 . . . 4 (𝑥 = 𝑤 → (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧)))
4541, 44anbi12d 747 . . 3 (𝑥 = 𝑤 → ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ (∀𝑦𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑤 → ∃𝑧𝐵 𝑦𝑅𝑧))))
4645rmo4 3432 . 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 224 1 (𝜑 → ∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 2030  wral 2941  wrex 2942  ∃*wrmo 2944   class class class wbr 4685   Or wor 5063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rmo 2949  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-po 5064  df-so 5065
This theorem is referenced by:  supexd  8400  supeu  8401
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