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Theorem sup00 8576
Description: The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
Assertion
Ref Expression
sup00 sup(𝐵, ∅, 𝑅) = ∅

Proof of Theorem sup00
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sup 8554 . 2 sup(𝐵, ∅, 𝑅) = {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 rab0 4119 . . 3 {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = ∅
32unieqi 4602 . 2 {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} =
4 uni0 4622 . 2 ∅ = ∅
51, 3, 43eqtri 2790 1 sup(𝐵, ∅, 𝑅) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wral 3054  wrex 3055  {crab 3058  c0 4078   cuni 4593   class class class wbr 4808  supcsup 8552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-dif 3734  df-in 3738  df-ss 3745  df-nul 4079  df-sn 4334  df-uni 4594  df-sup 8554
This theorem is referenced by:  inf00  8617
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