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Theorem sup0riota 8315
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 8305 . 2 (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧))))
3 ral0 4048 . . . . . 6 𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦
43biantrur 527 . . . . 5 (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)))
5 rex0 3914 . . . . . . 7 ¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧
6 imnot 355 . . . . . . 7 (¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥))
75, 6ax-mp 5 . . . . . 6 ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥)
87ralbii 2974 . . . . 5 (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥)
94, 8bitr3i 266 . . . 4 ((∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)) ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥)
109a1i 11 . . 3 (𝑅 Or 𝐴 → ((∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)) ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥))
1110riotabidv 6567 . 2 (𝑅 Or 𝐴 → (𝑥𝐴 (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧))) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
122, 11eqtrd 2655 1 (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wral 2907  wrex 2908  c0 3891   class class class wbr 4613   Or wor 4994  crio 6564  supcsup 8290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-po 4995  df-so 4996  df-iota 5810  df-riota 6565  df-sup 8292
This theorem is referenced by:  sup0  8316
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