Theorem dedekindle 11297

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 1195	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → 𝐴 ⊆ ℝ)
2		simpr2 1196	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → 𝐵 ⊆ ℝ)
3		simp1 1136	. . . . . . . . . 10 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → (𝐴 ∩ 𝐵) = ∅)
4		simpl 482	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐴)
5		disjel 4409	. . . . . . . . . 10 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝐵)
6	3, 4, 5	syl2an 596	. . . . . . . . 9 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ 𝑥 ∈ 𝐵)
7		eleq1w 2819	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
8	7	biimpcd 249	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 → (𝑦 = 𝑥 → 𝑥 ∈ 𝐵))
9	8	necon3bd 2946	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → (¬ 𝑥 ∈ 𝐵 → 𝑦 ≠ 𝑥))
10	9	ad2antll 729	. . . . . . . . 9 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (¬ 𝑥 ∈ 𝐵 → 𝑦 ≠ 𝑥))
11	6, 10	mpd 15	. . . . . . . 8 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ≠ 𝑥)
12		simp2 1137	. . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → 𝐴 ⊆ ℝ)
13		ssel2 3928	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
14	12, 4, 13	syl2an 596	. . . . . . . . . 10 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ ℝ)
15		simp3 1138	. . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → 𝐵 ⊆ ℝ)
16		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
17		ssel2 3928	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ ℝ ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ℝ)
18	15, 16, 17	syl2an 596	. . . . . . . . . 10 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ ℝ)
19	14, 18	ltlend 11278	. . . . . . . . 9 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥)))
20	19	biimprd 248	. . . . . . . 8 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥) → 𝑥 < 𝑦))
21	11, 20	mpan2d 694	. . . . . . 7 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≤ 𝑦 → 𝑥 < 𝑦))
22	21	ralimdvva 3183	. . . . . 6 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦))
23	22	3exp 1119	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ⊆ ℝ → (𝐵 ⊆ ℝ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦))))
24	23	3imp2 1350	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦)
25		dedekind 11296	. . . 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 1373	. . 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 412	. 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 4305	. . 3 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝐴 ∩ 𝐵))
29		simp1 1136	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → 𝐴 ⊆ ℝ)
30		elinel1 4153	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ 𝐴)
31		ssel2 3928	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
32	29, 30, 31	syl2an 596	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → 𝑤 ∈ ℝ)
33		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐴 ⊆ ℝ
34		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐵 ⊆ ℝ
35		nfra1 3260	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦
36	33, 34, 35	nf3an 1902	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)
37		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ∈ (𝐴 ∩ 𝐵)
38	36, 37	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑥((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵))
39		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝐴 ⊆ ℝ
40		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝐵 ⊆ ℝ
41		nfra2w 3272	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦
42	39, 40, 41	nf3an 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)
43		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)
44	42, 43	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴))
45		rsp 3224	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦))
46		elinel2 4154	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ 𝐵)
47		breq2 5102	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑤))
48	47	rspccv 3573	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑤 ∈ 𝐵 → 𝑥 ≤ 𝑤))
49	46, 48	syl5 34	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑥 ≤ 𝑤))
50	45, 49	syl6 35	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑥 ∈ 𝐴 → (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑥 ≤ 𝑤)))
51	50	com23 86	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑤 ∈ (𝐴 ∩ 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ≤ 𝑤)))
52	51	imp32 418	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) → 𝑥 ≤ 𝑤)
53	52	3ad2antl3 1188	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) → 𝑥 ≤ 𝑤)
54	53	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑥 ≤ 𝑤)
55		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)
56	30	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑤 ∈ 𝐴)
57		breq1 5101	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
58	57	ralbidv 3159	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑤 ≤ 𝑦))
59	58	rspccva 3575	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ∧ 𝑤 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 𝑤 ≤ 𝑦)
60	55, 56, 59	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) → ∀𝑦 ∈ 𝐵 𝑤 ≤ 𝑦)
61	60	r19.21bi 3228	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑤 ≤ 𝑦)
62	54, 61	jca 511	. . . . . . . . . 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 412	. . . . . . . . 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 3234	. . . . . . . 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 456	. . . . . . 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 3234	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦))
67		breq2 5102	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑤))
68		breq1 5101	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
69	67, 68	anbi12d 632	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) ↔ (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦)))
70	69	2ralbidv 3200	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦)))
71	70	rspcev 3576	. . . . . 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 584	. . . . 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 413	. . . 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 1931	. . 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 217	. 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 3015	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285   class class class wbr 5098  ℝcr 11025   < clt 11166   ≤ cle 11167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-mulcl 11088  ax-mulrcl 11089  ax-i2m1 11094  ax-1ne0 11095  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172

This theorem is referenced by:  axcontlem10  29046