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Theorem sup00 8367
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 8345 . 2 sup(𝐵, ∅, 𝑅) = {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 rab0 3953 . . 3 {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = ∅
32unieqi 4443 . 2 {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} =
4 uni0 4463 . 2 ∅ = ∅
51, 3, 43eqtri 2647 1 sup(𝐵, ∅, 𝑅) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1482  wral 2911  wrex 2912  {crab 2915  c0 3913   cuni 4434   class class class wbr 4651  supcsup 8343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-in 3579  df-ss 3586  df-nul 3914  df-sn 4176  df-uni 4435  df-sup 8345
This theorem is referenced by:  inf00  8408
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