Theorem supmo 9141

Description: Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by NM, 5-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
supmo.1	⊢ (𝜑 → 𝑅 Or 𝐴)

Assertion

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

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

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem supmo

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supmo.1	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐴)
2		ancom 460	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦))
3	2	anbi2ci 624	. . . . . . 7 ⊢ (((∀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)))
4		an42 653	. . . . . . 7 ⊢ (((∀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))))
5		an42 653	. . . . . . 7 ⊢ (((∀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)))
6	3, 4, 5	3bitr4i 302	. . . . . 6 ⊢ (((∀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)))
7		ralnex 3163	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ¬ ∃𝑦 ∈ 𝐵 𝑥𝑅𝑦)
8		breq1 5073	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑤 ↔ 𝑥𝑅𝑤))
9		breq1 5073	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑥𝑅𝑧))
10	9	rexbidv 3225	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (∃𝑧 ∈ 𝐵 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑥𝑅𝑧))
11	8, 10	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑥𝑅𝑧)))
12	11	rspcva 3550	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (𝑥𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑥𝑅𝑧))
13		breq2 5074	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑧))
14	13	cbvrexvw 3373	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝐵 𝑥𝑅𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑥𝑅𝑧)
15	12, 14	syl6ibr 251	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (𝑥𝑅𝑤 → ∃𝑦 ∈ 𝐵 𝑥𝑅𝑦))
16	15	con3d 152	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (¬ ∃𝑦 ∈ 𝐵 𝑥𝑅𝑦 → ¬ 𝑥𝑅𝑤))
17	7, 16	syl5bi 241	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 → ¬ 𝑥𝑅𝑤))
18	17	expimpd 453	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦) → ¬ 𝑥𝑅𝑤))
19	18	ad2antrl 724	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦) → ¬ 𝑥𝑅𝑤))
20		ralnex 3163	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ↔ ¬ ∃𝑦 ∈ 𝐵 𝑤𝑅𝑦)
21		breq1 5073	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝑦𝑅𝑥 ↔ 𝑤𝑅𝑥))
22		breq1 5073	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → (𝑦𝑅𝑧 ↔ 𝑤𝑅𝑧))
23	22	rexbidv 3225	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (∃𝑧 ∈ 𝐵 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
24	21, 23	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
25	24	rspcva 3550	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
26		breq2 5074	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑤𝑅𝑦 ↔ 𝑤𝑅𝑧))
27	26	cbvrexvw 3373	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝐵 𝑤𝑅𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)
28	25, 27	syl6ibr 251	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (𝑤𝑅𝑥 → ∃𝑦 ∈ 𝐵 𝑤𝑅𝑦))
29	28	con3d 152	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (¬ ∃𝑦 ∈ 𝐵 𝑤𝑅𝑦 → ¬ 𝑤𝑅𝑥))
30	20, 29	syl5bi 241	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 → ¬ 𝑤𝑅𝑥))
31	30	expimpd 453	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦) → ¬ 𝑤𝑅𝑥))
32	31	ad2antll 725	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦) → ¬ 𝑤𝑅𝑥))
33	19, 32	anim12d 608	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦) ∧ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦)) → (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
34	6, 33	syl5bi 241	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
35		sotrieq2 5524	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑥 = 𝑤 ↔ (¬ 𝑥𝑅𝑤 ∧ ¬ 𝑤𝑅𝑥)))
36	34, 35	sylibrd 258	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → 𝑥 = 𝑤))
37	36	ralrimivva 3114	. . 3 ⊢ (𝑅 Or 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → 𝑥 = 𝑤))
38	1, 37	syl 17	. 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))
39		breq1 5073	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥𝑅𝑦 ↔ 𝑤𝑅𝑦))
40	39	notbid 317	. . . . 5 ⊢ (𝑥 = 𝑤 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑤𝑅𝑦))
41	40	ralbidv 3120	. . . 4 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦))
42		breq2 5074	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑤))
43	42	imbi1d 341	. . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
44	43	ralbidv 3120	. . . 4 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
45	41, 44	anbi12d 630	. . 3 ⊢ (𝑥 = 𝑤 → ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝑤𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑤 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
46	45	rmo4 3660	. 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))
47	38, 46	sylibr 233	1 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  ∃*wrmo 3066   class class class wbr 5070   Or wor 5493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rmo 3071  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-po 5494  df-so 5495

This theorem is referenced by:  supexd  9142  supeu  9143