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Theorem sup0riota 8775
 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 8765 . 2 (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧))))
3 ral0 4370 . . . . . 6 𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦
43biantrur 531 . . . . 5 (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)))
5 rex0 4237 . . . . . . 7 ¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧
6 imnot 367 . . . . . . 7 (¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥))
75, 6ax-mp 5 . . . . . 6 ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥)
87ralbii 3132 . . . . 5 (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥)
94, 8bitr3i 278 . . . 4 ((∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)) ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥)
109a1i 11 . . 3 (𝑅 Or 𝐴 → ((∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)) ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥))
1110riotabidv 6979 . 2 (𝑅 Or 𝐴 → (𝑥𝐴 (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧))) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
122, 11eqtrd 2831 1 (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522  ∀wral 3105  ∃wrex 3106  ∅c0 4211   class class class wbr 4962   Or wor 5361  ℩crio 6976  supcsup 8750 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-po 5362  df-so 5363  df-iota 6189  df-riota 6977  df-sup 8752 This theorem is referenced by:  sup0  8776
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