Theorem sup00 6968

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 6949	. 2 ⊢ sup(𝐵, ∅, 𝑅) = ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}
2		rab0 3437	. . 3 ⊢ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∅
3	2	unieqi 3799	. 2 ⊢ ∪ {𝑥 ∈ ∅ ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ ∅ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∪ ∅
4		uni0 3816	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2190	1 ⊢ sup(𝐵, ∅, 𝑅) = ∅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1343  ∀wral 2444  ∃wrex 2445  {crab 2448  ∅c0 3409  ∪ cuni 3789   class class class wbr 3982  supcsup 6947

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-uni 3790  df-sup 6949

This theorem is referenced by:  inf00  6996