Theorem dedekindle 11308

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 1201	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → 𝐴 ⊆ ℝ)
2		simpr2 1202	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → 𝐵 ⊆ ℝ)
3		simp1 1142	. . . . . . . . . 10 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → (𝐴 ∩ 𝐵) = ∅)
4		simpl 483	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐴)
5		disjel 4392	. . . . . . . . . 10 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝐵)
6	3, 4, 5	syl2an 602	. . . . . . . . 9 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ 𝑥 ∈ 𝐵)
7		eleq1w 2823	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
8	7	biimpcd 250	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 → (𝑦 = 𝑥 → 𝑥 ∈ 𝐵))
9	8	necon3bd 2949	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → (¬ 𝑥 ∈ 𝐵 → 𝑦 ≠ 𝑥))
10	9	ad2antll 735	. . . . . . . . 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 1143	. . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → 𝐴 ⊆ ℝ)
13		ssel2 3917	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
14	12, 4, 13	syl2an 602	. . . . . . . . . 10 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ ℝ)
15		simp3 1144	. . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → 𝐵 ⊆ ℝ)
16		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
17		ssel2 3917	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ ℝ ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ℝ)
18	15, 16, 17	syl2an 602	. . . . . . . . . 10 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ ℝ)
19	14, 18	ltlend 11289	. . . . . . . . 9 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 < 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥)))
20	19	biimprd 249	. . . . . . . 8 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≠ 𝑥) → 𝑥 < 𝑦))
21	11, 20	mpan2d 700	. . . . . . 7 ⊢ ((((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≤ 𝑦 → 𝑥 < 𝑦))
22	21	ralimdvva 3187	. . . . . 6 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦))
23	22	3exp 1125	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ⊆ ℝ → (𝐵 ⊆ ℝ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦))))
24	23	3imp2 1356	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦)
25		dedekind 11307	. . . 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 1379	. . 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 413	. 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 4288	. . 3 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝐴 ∩ 𝐵))
29		simp1 1142	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → 𝐴 ⊆ ℝ)
30		elinel1 4137	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ 𝐴)
31		ssel2 3917	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
32	29, 30, 31	syl2an 602	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → 𝑤 ∈ ℝ)
33		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐴 ⊆ ℝ
34		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐵 ⊆ ℝ
35		nfra1 3264	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦
36	33, 34, 35	nf3an 1908	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)
37		nfv 1921	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ∈ (𝐴 ∩ 𝐵)
38	36, 37	nfan 1906	. . . . . . 7 ⊢ Ⅎ𝑥((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵))
39		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝐴 ⊆ ℝ
40		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝐵 ⊆ ℝ
41		nfra2w 3276	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦
42	39, 40, 41	nf3an 1908	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)
43		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)
44	42, 43	nfan 1906	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴))
45		rsp 3228	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦))
46		elinel2 4138	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ 𝐵)
47		breq2 5083	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑤))
48	47	rspccv 3564	. . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) → 𝑥 ≤ 𝑤)
53	52	3ad2antl3 1194	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) → 𝑥 ≤ 𝑤)
54	53	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑥 ≤ 𝑤)
55		simp3 1144	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)
56	30	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑤 ∈ 𝐴)
57		breq1 5082	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
58	57	ralbidv 3163	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑤 ≤ 𝑦))
59	58	rspccva 3566	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ∧ 𝑤 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 𝑤 ≤ 𝑦)
60	55, 56, 59	syl2an 602	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) → ∀𝑦 ∈ 𝐵 𝑤 ≤ 𝑦)
61	60	r19.21bi 3232	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑤 ≤ 𝑦)
62	54, 61	jca 516	. . . . . . . . . 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 413	. . . . . . . . 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 3238	. . . . . . . 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 457	. . . . . . 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 3238	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦))
67		breq2 5083	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑤))
68		breq1 5082	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
69	67, 68	anbi12d 638	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) ↔ (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦)))
70	69	2ralbidv 3204	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦)))
71	70	rspcev 3567	. . . . . 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 590	. . . . 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 414	. . . 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 1937	. . 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 218	. 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 3018	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 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∃wrex 3064   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268   class class class wbr 5079  ℝcr 11035   < clt 11177   ≤ cle 11178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-mulcl 11098  ax-mulrcl 11099  ax-i2m1 11104  ax-1ne0 11105  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7366  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183

This theorem is referenced by:  axcontlem10  29067