Theorem sup00 6883

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 6864	. 2 ⊢ sup(𝐵, ∅, 𝑅) = ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}
2		rab0 3386	. . 3 ⊢ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∅
3	2	unieqi 3741	. 2 ⊢ ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∪ ∅
4		uni0 3758	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2162	1 ⊢ sup(𝐵, ∅, 𝑅) = ∅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331  ∀wral 2414  ∃wrex 2415  {crab 2418  ∅c0 3358  ∪ cuni 3731   class class class wbr 3924  supcsup 6862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528  df-uni 3732  df-sup 6864

This theorem is referenced by:  inf00  6911