MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sup0riota Structured version   Visualization version   GIF version

Theorem sup0riota 9503
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 9493 . 2 (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧))))
3 ral0 4519 . . . . . 6 𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦
43biantrur 530 . . . . 5 (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)))
5 rex0 4366 . . . . . . 7 ¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧
6 imnot 365 . . . . . . 7 (¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥))
75, 6ax-mp 5 . . . . . 6 ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥)
87ralbii 3091 . . . . 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 2775 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 1537  wral 3059  wrex 3068  c0 4339   class class class wbr 5148   Or wor 5596  crio 7387  supcsup 9478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-po 5597  df-so 5598  df-iota 6516  df-riota 7388  df-sup 9480
This theorem is referenced by:  sup0  9504
  Copyright terms: Public domain W3C validator