Theorem dedekindle 11385

Description: The Dedekind cut theorem, with the hypothesis weakened to only require non-strict less than. (Contributed by Scott Fenton, 2-Jul-2013.)

Assertion

Ref	Expression
dedekindle	⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))

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

Proof of Theorem dedekindle

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr1 1213	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → 𝐴 ⊆ ℝ)
2		simpr2 1214	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → 𝐵 ⊆ ℝ)
3		simp1 1154	. . . . . . . . . 10 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → (𝐴 ∩ 𝐵) = ∅)
4		simpl 488	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐴)
5		disjel 4417	. . . . . . . . . 10 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝐵)
6	3, 4, 5	syl2an 608	. . . . . . . . 9 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ 𝑥 ∈ 𝐵)
7		eleq1w 2848	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
8	7	biimpcd 252	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 → (𝑦 = 𝑥 → 𝑥 ∈ 𝐵))
9	8	necon3bd 2974	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → (¬ 𝑥 ∈ 𝐵 → 𝑦 ≠ 𝑥))
10	9	ad2antll 742	. . . . . . . . 9 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (¬ 𝑥 ∈ 𝐵 → 𝑦 ≠ 𝑥))
11	6, 10	mpd 16	. . . . . . . 8 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ≠ 𝑥)
12		simp2 1155	. . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → 𝐴 ⊆ ℝ)
13		ssel2 3933	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
14	12, 4, 13	syl2an 608	. . . . . . . . . 10 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ ℝ)
15		simp3 1156	. . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → 𝐵 ⊆ ℝ)
16		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
17		ssel2 3933	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ ℝ ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ℝ)
18	15, 16, 17	syl2an 608	. . . . . . . . . 10 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ ℝ)
19	14, 18	ltlend 11366	. . . . . . . . 9 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥)))
20	19	biimprd 251	. . . . . . . 8 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥) → 𝑥 < 𝑦))
21	11, 20	mpan2d 707	. . . . . . 7 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≤ 𝑦 → 𝑥 < 𝑦))
22	21	ralimdvva 3214	. . . . . 6 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦))
23	22	3exp 1137	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ⊆ ℝ → (𝐵 ⊆ ℝ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦))))
24	23	3imp2 1368	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦)
25		dedekind 11384	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))
26	1, 2, 24, 25	syl3anc 1398	. . 3 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))
27	26	ex 418	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)))
28		n0 4307	. . 3 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝐴 ∩ 𝐵))
29		simp1 1154	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → 𝐴 ⊆ ℝ)
30		elinel1 4154	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ 𝐴)
31		ssel2 3933	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
32	29, 30, 31	syl2an 608	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → 𝑤 ∈ ℝ)
33		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐴 ⊆ ℝ
34		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐵 ⊆ ℝ
35		nfra1 3291	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦
36	33, 34, 35	nf3an 1934	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)
37		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ∈ (𝐴 ∩ 𝐵)
38	36, 37	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑥((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵))
39		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝐴 ⊆ ℝ
40		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝐵 ⊆ ℝ
41		nfra2w 3303	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦
42	39, 40, 41	nf3an 1934	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)
43		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)
44	42, 43	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴))
45		rsp 3255	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦))
46		elinel2 4155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ 𝐵)
47		breq2 5115	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑤))
48	47	rspccv 3580	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑤 ∈ 𝐵 → 𝑥 ≤ 𝑤))
49	46, 48	syl5 35	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑥 ≤ 𝑤))
50	45, 49	syl6 36	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑥 ∈ 𝐴 → (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑥 ≤ 𝑤)))
51	50	com23 87	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑤 ∈ (𝐴 ∩ 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ≤ 𝑤)))
52	51	imp32 424	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) → 𝑥 ≤ 𝑤)
53	52	3ad2antl3 1206	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) → 𝑥 ≤ 𝑤)
54	53	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑥 ≤ 𝑤)
55		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)
56	30	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑤 ∈ 𝐴)
57		breq1 5114	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
58	57	ralbidv 3190	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑤 ≤ 𝑦))
59	58	rspccva 3582	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ∧ 𝑤 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 𝑤 ≤ 𝑦)
60	55, 56, 59	syl2an 608	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) → ∀𝑦 ∈ 𝐵 𝑤 ≤ 𝑦)
61	60	r19.21bi 3259	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑤 ≤ 𝑦)
62	54, 61	jca 521	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦))
63	62	ex 418	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) → (𝑦 ∈ 𝐵 → (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦)))
64	44, 63	ralrimi 3265	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) → ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦))
65	64	expr 462	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦)))
66	38, 65	ralrimi 3265	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦))
67		breq2 5115	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑤))
68		breq1 5114	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
69	67, 68	anbi12d 644	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) ↔ (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦)))
70	69	2ralbidv 3231	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦)))
71	70	rspcev 3583	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦)) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))
72	32, 66, 71	syl2anc 596	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))
73	72	expcom 419	. . . 4 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)))
74	73	exlimiv 1963	. . 3 ⊢ (∃𝑤 𝑤 ∈ (𝐴 ∩ 𝐵) → ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)))
75	28, 74	sylbi 220	. 2 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ → ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)))
76	27, 75	pm2.61ine 3043	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286   class class class wbr 5111  ℝcr 11110   < clt 11254   ≤ cle 11255

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-mulcl 11173  ax-mulrcl 11174  ax-i2m1 11179  ax-1ne0 11180  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-sup 11189

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-po 5571  df-so 5572  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260

This theorem is used by:  axcontlem10  29352