Theorem dedekindle 11426

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 1194	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → 𝐴 ⊆ ℝ)
2		simpr2 1195	. . . 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 4456	. . . . . . . . . 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 2823	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
8	7	biimpcd 249	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 → (𝑦 = 𝑥 → 𝑥 ∈ 𝐵))
9	8	necon3bd 2953	. . . . . . . . . 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 3977	. . . . . . . . . . 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 3977	. . . . . . . . . . 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 11407	. . . . . . . . 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 3205	. . . . . 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 1349	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦)
25		dedekind 11425	. . . 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 1372	. . 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 4352	. . 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 4200	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ 𝐴)
31		ssel2 3977	. . . . . . 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 1913	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐴 ⊆ ℝ
34		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐵 ⊆ ℝ
35		nfra1 3283	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦
36	33, 34, 35	nf3an 1900	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)
37		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ∈ (𝐴 ∩ 𝐵)
38	36, 37	nfan 1898	. . . . . . 7 ⊢ Ⅎ𝑥((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵))
39		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝐴 ⊆ ℝ
40		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝐵 ⊆ ℝ
41		nfra2w 3298	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦
42	39, 40, 41	nf3an 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦)
43		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴)
44	42, 43	nfan 1898	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ (𝑤 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐴))
45		rsp 3246	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦))
46		elinel2 4201	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (𝐴 ∩ 𝐵) → 𝑤 ∈ 𝐵)
47		breq2 5146	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑤))
48	47	rspccv 3618	. . . . . . . . . . . . . . . . 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 1187	. . . . . . . . . . . 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 5145	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦))
58	57	ralbidv 3177	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑤 ≤ 𝑦))
59	58	rspccva 3620	. . . . . . . . . . . . 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 3250	. . . . . . . . . . 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 3256	. . . . . . . 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 3256	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ∧ 𝑤 ∈ (𝐴 ∩ 𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦))
67		breq2 5146	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑤))
68		breq1 5145	. . . . . . . . 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 3220	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑦)))
71	70	rspcev 3621	. . . . . 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 1929	. . 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 3024	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 1539  ∃wex 1778   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332   class class class wbr 5142  ℝcr 11155   < clt 11296   ≤ cle 11297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-mulcl 11218  ax-mulrcl 11219  ax-i2m1 11224  ax-1ne0 11225  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-sup 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302

This theorem is referenced by:  axcontlem10  28989