Theorem sup00 7245

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.

Step	Hyp	Ref	Expression
1		df-sup 7226	. 2 ⊢ sup(𝐵, ∅, 𝑅) = ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}
2		rab0 3525	. . 3 ⊢ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∅
3	2	unieqi 3908	. 2 ⊢ ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∪ ∅
4		uni0 3925	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2256	1 ⊢ sup(𝐵, ∅, 𝑅) = ∅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1398  ∀wral 2511  ∃wrex 2512  {crab 2515  ∅c0 3496  ∪ cuni 3898   class class class wbr 4093  supcsup 7224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497  df-sn 3679  df-uni 3899  df-sup 7226

This theorem is referenced by:  inf00  7273