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Theorem sup00 8535
 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 8513 . 2 sup(𝐵, ∅, 𝑅) = {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 rab0 4098 . . 3 {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = ∅
32unieqi 4597 . 2 {𝑥 ∈ ∅ ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} =
4 uni0 4617 . 2 ∅ = ∅
51, 3, 43eqtri 2786 1 sup(𝐵, ∅, 𝑅) = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1632  ∀wral 3050  ∃wrex 3051  {crab 3054  ∅c0 4058  ∪ cuni 4588   class class class wbr 4804  supcsup 8511 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322  df-uni 4589  df-sup 8513 This theorem is referenced by:  inf00  8576
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