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Theorem sup00 6407
 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 6390 . 2 sup(𝐵, ∅, 𝑅) = {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 rab0 3274 . . 3 {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = ∅
32unieqi 3618 . 2 {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} =
4 uni0 3635 . 2 ∅ = ∅
51, 3, 43eqtri 2080 1 sup(𝐵, ∅, 𝑅) = ∅
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   = wceq 1259  ∀wral 2323  ∃wrex 2324  {crab 2327  ∅c0 3252  ∪ cuni 3608   class class class wbr 3792  supcsup 6388 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-in 2952  df-ss 2959  df-nul 3253  df-sn 3409  df-uni 3609  df-sup 6390 This theorem is referenced by: (None)
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