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Theorem sup00 9456
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 9434 . 2 sup(𝐵, ∅, 𝑅) = {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 rab0 4375 . . 3 {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = ∅
32unieqi 4912 . 2 {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} =
4 uni0 4930 . 2 ∅ = ∅
51, 3, 43eqtri 2756 1 sup(𝐵, ∅, 𝑅) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wral 3053  wrex 3062  {crab 3424  c0 4315   cuni 4900   class class class wbr 5139  supcsup 9432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958  df-nul 4316  df-sn 4622  df-uni 4901  df-sup 9434
This theorem is referenced by:  inf00  9498
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