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Theorem sup0riota 9505
Description: The supremum of an empty set is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
Assertion
Ref Expression
sup0riota (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦

Proof of Theorem sup0riota
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝑅 Or 𝐴𝑅 Or 𝐴)
21supval2 9495 . 2 (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧))))
3 ral0 4513 . . . . . 6 𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦
43biantrur 530 . . . . 5 (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)))
5 rex0 4360 . . . . . . 7 ¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧
6 imnot 365 . . . . . . 7 (¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥))
75, 6ax-mp 5 . . . . . 6 ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥)
87ralbii 3093 . . . . 5 (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥)
94, 8bitr3i 277 . . . 4 ((∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)) ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥)
109a1i 11 . . 3 (𝑅 Or 𝐴 → ((∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)) ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥))
1110riotabidv 7390 . 2 (𝑅 Or 𝐴 → (𝑥𝐴 (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧))) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
122, 11eqtrd 2777 1 (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wral 3061  wrex 3070  c0 4333   class class class wbr 5143   Or wor 5591  crio 7387  supcsup 9480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-po 5592  df-so 5593  df-iota 6514  df-riota 7388  df-sup 9482
This theorem is referenced by:  sup0  9506
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