Theorem supmoti 6986

Description: Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 8027) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.)

Hypothesis

Ref	Expression
supmoti.ti	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))

Assertion

Ref	Expression
supmoti	⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))

Distinct variable groups:   𝑢,𝐴,𝑣,𝑥   𝑦,𝐴,𝑥,𝑧   𝑥,𝐵,𝑦,𝑧   𝑢,𝑅,𝑣,𝑥   𝑦,𝑅,𝑧   𝜑,𝑢,𝑣,𝑥

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑣,𝑢)

Proof of Theorem supmoti

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ancom 266	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦))
2	1	anbi2ci 459	. . . . . 6 ⊢ (((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦) ∧ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ↔ ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦)))
3		an42 587	. . . . . 6 ⊢ (((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ↔ ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦) ∧ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
4		an42 587	. . . . . 6 ⊢ (((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦) ∧ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦)) ↔ ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦)))
5	2, 3, 4	3bitr4i 212	. . . . 5 ⊢ (((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ↔ ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦) ∧ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦)))
6		ralnex 2465	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ¬ ∃𝑦 ∈ 𝐵 𝑥𝑅𝑦)
7		breq1 4003	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑤 ↔ 𝑥𝑅𝑤))
8		breq1 4003	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑥𝑅𝑧))
9	8	rexbidv 2478	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (∃𝑧 ∈ 𝐵 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑥𝑅𝑧))
10	7, 9	imbi12d 234	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑥𝑅𝑧)))
11	10	rspcva 2839	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (𝑥𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑥𝑅𝑧))
12		breq2 4004	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑧))
13	12	cbvrexv 2704	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐵 𝑥𝑅𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑥𝑅𝑧)
14	11, 13	syl6ibr 162	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (𝑥𝑅𝑤 → ∃𝑦 ∈ 𝐵 𝑥𝑅𝑦))
15	14	con3d 631	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (¬ ∃𝑦 ∈ 𝐵 𝑥𝑅𝑦 → ¬ 𝑥𝑅𝑤))
16	6, 15	biimtrid 152	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 → ¬ 𝑥𝑅𝑤))
17	16	expimpd 363	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦) → ¬ 𝑥𝑅𝑤))
18	17	ad2antrl 490	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦) → ¬ 𝑥𝑅𝑤))
19		ralnex 2465	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ↔ ¬ ∃𝑦 ∈ 𝐵 𝑤𝑅𝑦)
20		breq1 4003	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑦𝑅𝑥 ↔ 𝑤𝑅𝑥))
21		breq1 4003	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝑦𝑅𝑧 ↔ 𝑤𝑅𝑧))
22	21	rexbidv 2478	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (∃𝑧 ∈ 𝐵 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
23	20, 22	imbi12d 234	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
24	23	rspcva 2839	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
25		breq2 4004	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑤𝑅𝑦 ↔ 𝑤𝑅𝑧))
26	25	cbvrexv 2704	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐵 𝑤𝑅𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)
27	24, 26	syl6ibr 162	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (𝑤𝑅𝑥 → ∃𝑦 ∈ 𝐵 𝑤𝑅𝑦))
28	27	con3d 631	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (¬ ∃𝑦 ∈ 𝐵 𝑤𝑅𝑦 → ¬ 𝑤𝑅𝑥))
29	19, 28	biimtrid 152	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 → ¬ 𝑤𝑅𝑥))
30	29	expimpd 363	. . . . . . 7 ⊢ (𝑤 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦) → ¬ 𝑤𝑅𝑥))
31	30	ad2antll 491	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦) → ¬ 𝑤𝑅𝑥))
32	18, 31	anim12d 335	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦) ∧ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦)) → (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
33	5, 32	biimtrid 152	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
34		supmoti.ti	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
35	34	ralrimivva 2559	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
36		equequ1 1712	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑢 = 𝑣 ↔ 𝑥 = 𝑣))
37		breq1 4003	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (𝑢𝑅𝑣 ↔ 𝑥𝑅𝑣))
38	37	notbid 667	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑥𝑅𝑣))
39		breq2 4004	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (𝑣𝑅𝑢 ↔ 𝑣𝑅𝑥))
40	39	notbid 667	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (¬ 𝑣𝑅𝑢 ↔ ¬ 𝑣𝑅𝑥))
41	38, 40	anbi12d 473	. . . . . . 7 ⊢ (𝑢 = 𝑥 → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝑥𝑅𝑣 ∧ ¬ 𝑣𝑅𝑥)))
42	36, 41	bibi12d 235	. . . . . 6 ⊢ (𝑢 = 𝑥 → ((𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ (𝑥 = 𝑣 ↔ (¬ 𝑥𝑅𝑣 ∧ ¬ 𝑣𝑅𝑥))))
43		equequ2 1713	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑥 = 𝑣 ↔ 𝑥 = 𝑤))
44		breq2 4004	. . . . . . . . 9 ⊢ (𝑣 = 𝑤 → (𝑥𝑅𝑣 ↔ 𝑥𝑅𝑤))
45	44	notbid 667	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (¬ 𝑥𝑅𝑣 ↔ ¬ 𝑥𝑅𝑤))
46		breq1 4003	. . . . . . . . 9 ⊢ (𝑣 = 𝑤 → (𝑣𝑅𝑥 ↔ 𝑤𝑅𝑥))
47	46	notbid 667	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (¬ 𝑣𝑅𝑥 ↔ ¬ 𝑤𝑅𝑥))
48	45, 47	anbi12d 473	. . . . . . 7 ⊢ (𝑣 = 𝑤 → ((¬ 𝑥𝑅𝑣 ∧ ¬ 𝑣𝑅𝑥) ↔ (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
49	43, 48	bibi12d 235	. . . . . 6 ⊢ (𝑣 = 𝑤 → ((𝑥 = 𝑣 ↔ (¬ 𝑥𝑅𝑣 ∧ ¬ 𝑣𝑅𝑥)) ↔ (𝑥 = 𝑤 ↔ (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥))))
50	42, 49	rspc2v 2854	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → (𝑥 = 𝑤 ↔ (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥))))
51	35, 50	mpan9 281	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑥 = 𝑤 ↔ (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
52	33, 51	sylibrd 169	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → 𝑥 = 𝑤))
53	52	ralrimivva 2559	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → 𝑥 = 𝑤))
54		breq1 4003	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥𝑅𝑦 ↔ 𝑤𝑅𝑦))
55	54	notbid 667	. . . . 5 ⊢ (𝑥 = 𝑤 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
56	55	ralbidv 2477	. . . 4 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦))
57		breq2 4004	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑤))
58	57	imbi1d 231	. . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
59	58	ralbidv 2477	. . . 4 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
60	56, 59	anbi12d 473	. . 3 ⊢ (𝑥 = 𝑤 → ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
61	60	rmo4 2930	. 2 ⊢ (∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → 𝑥 = 𝑤))
62	53, 61	sylibr 134	1 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  ∃*wrmo 2458   class class class wbr 4000

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rmo 2463  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001

This theorem is referenced by:  supeuti  6987  infmoti  7021