Definition df-sup 9434

Description: Define the supremum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the supremum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrtval 15181. See dfsup2 9436 for alternate definition not requiring dummy variables. (Contributed by NM, 22-May-1999.)

Assertion

Ref	Expression
df-sup	⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧

Detailed syntax breakdown of Definition df-sup

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3		cR	. . 3 class 𝑅
4	1, 2, 3	csup 9432	. 2 class sup(𝐴, 𝐵, 𝑅)
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1541	. . . . . . . 8 class 𝑥
7		vy	. . . . . . . . 9 setvar 𝑦
8	7	cv 1541	. . . . . . . 8 class 𝑦
9	6, 8, 3	wbr 5148	. . . . . . 7 wff 𝑥𝑅𝑦
10	9	wn 3	. . . . . 6 wff ¬ 𝑥𝑅𝑦
11	10, 7, 1	wral 3062	. . . . 5 wff ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦
12	8, 6, 3	wbr 5148	. . . . . . 7 wff 𝑦𝑅𝑥
13		vz	. . . . . . . . . 10 setvar 𝑧
14	13	cv 1541	. . . . . . . . 9 class 𝑧
15	8, 14, 3	wbr 5148	. . . . . . . 8 wff 𝑦𝑅𝑧
16	15, 13, 1	wrex 3071	. . . . . . 7 wff ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧
17	12, 16	wi 4	. . . . . 6 wff (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)
18	17, 7, 2	wral 3062	. . . . 5 wff ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)
19	11, 18	wa 397	. . . 4 wff (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))
20	19, 5, 2	crab 3433	. . 3 class {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
21	20	cuni 4908	. 2 class ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
22	4, 21	wceq 1542	1 wff sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}

This definition is referenced by:  dfsup2  9436  supeq1  9437  supeq2  9440  supeq3  9441  supeq123d  9442  supexd  9445  supval2  9447  sup00  9456  dfinfre  12192  prdsds  17407  supex2g  36594