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Theorem ixxlb 13389

Description: Extract the lower bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by AV, 12-Sep-2020.)

**Hypotheses**
Ref	Expression
ixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
ixxub.2	⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤𝑆𝐵))
ixxub.3	⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤𝑆𝐵 → 𝑤 ≤ 𝐵))
ixxub.4	⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴𝑅𝑤))
ixxub.5	⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴𝑅𝑤 → 𝐴 ≤ 𝑤))

**Assertion**
Ref	Expression
ixxlb	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^*, < ) = 𝐴)

Distinct variable groups: 𝑥,𝑤,𝑦,𝑧,𝐴 𝑤,𝑂 𝑤,𝐵,𝑥,𝑦,𝑧 𝑥,𝑅,𝑦,𝑧 𝑥,𝑆,𝑦,𝑧 Allowed substitution hints: 𝑅(𝑤) 𝑆(𝑤) 𝑂(𝑥,𝑦,𝑧)

**Proof of Theorem ixxlb**
Step	Hyp	Ref	Expression
1		ixx.1	. . . . . . . . 9 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
2	1	elixx1 13376	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵)))
3	2	3adant3 1150	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵)))
4	3	biimpa 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵))
5	4	simp1d 1160	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝑤 ∈ ℝ^*)
6	5	ex 417	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ ℝ^*))
7	6	ssrdv 3943	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴𝑂𝐵) ⊆ ℝ^*)
8		infxrcl 13355	. . 3 ⊢ ((𝐴𝑂𝐵) ⊆ ℝ^* → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^)
9	7, 8	syl 18	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^)
10		simp1 1154	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → 𝐴 ∈ ℝ^*)
11		simprr 784	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))
12	7	ad2antrr 738	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → (𝐴𝑂𝐵) ⊆ ℝ^)
13		qre 12972	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ℚ → 𝑤 ∈ ℝ)
14	13	rexrd 11254	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℚ → 𝑤 ∈ ℝ^*)
15	14	ad2antlr 739	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → 𝑤 ∈ ℝ^)
16		simprl 782	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝐴 < 𝑤)
17	10	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → 𝐴 ∈ ℝ^)
18		ixxub.4	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴𝑅𝑤))
19	17, 15, 18	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → (𝐴 < 𝑤 → 𝐴𝑅𝑤))
20	16, 19	mpd 16	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝐴𝑅𝑤)
21	9	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^*)
22		simpll2 1232	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → 𝐵 ∈ ℝ^)
23		simp3 1156	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴𝑂𝐵) ≠ ∅)
24		n0 4307	. . . . . . . . . . . . . 14 ⊢ ((𝐴𝑂𝐵) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝐴𝑂𝐵))
25	23, 24	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ∃𝑤 𝑤 ∈ (𝐴𝑂𝐵))
26	9	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^)
27		simpl2 1211	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝐵 ∈ ℝ^*)
28		infxrlb 13356	. . . . . . . . . . . . . . 15 ⊢ (((𝐴𝑂𝐵) ⊆ ℝ^* ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝑤)
29	7, 28	sylan 591	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝑤)
30	4	simp3d 1162	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝑤𝑆𝐵)
31		ixxub.3	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤𝑆𝐵 → 𝑤 ≤ 𝐵))
32	5, 27, 31	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → (𝑤𝑆𝐵 → 𝑤 ≤ 𝐵))
33	30, 32	mpd 16	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝑤 ≤ 𝐵)
34	26, 5, 27, 29, 33	xrletrd 13182	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝐵)
35	25, 34	exlimddv 1965	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝐵)
36	35	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → inf((𝐴𝑂𝐵), ℝ^, < ) ≤ 𝐵)
37	15, 21, 22, 11, 36	xrltletrd 13181	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝑤 < 𝐵)
38		ixxub.2	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤𝑆𝐵))
39	15, 22, 38	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → (𝑤 < 𝐵 → 𝑤𝑆𝐵))
40	37, 39	mpd 16	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝑤𝑆𝐵)
41	3	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵)))
42	15, 20, 40, 41	mpbir3and 1361	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝑤 ∈ (𝐴𝑂𝐵))
43	12, 42, 28	syl2anc 595	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → inf((𝐴𝑂𝐵), ℝ^, < ) ≤ 𝑤)
44	21, 15	xrlenltd 11270	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → (inf((𝐴𝑂𝐵), ℝ^, < ) ≤ 𝑤 ↔ ¬ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < )))
45	43, 44	mpbid 235	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → ¬ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))
46	11, 45	pm2.65da 828	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) → ¬ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < )))
47	46	nrexdv 3160	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ¬ ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < )))
48		qbtwnxr 13221	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^ ∧ 𝐴 < inf((𝐴𝑂𝐵), ℝ^, < )) → ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < )))
49	48	3expia 1139	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^) → (𝐴 < inf((𝐴𝑂𝐵), ℝ^, < ) → ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))))
50	10, 9, 49	syl2anc 595	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴 < inf((𝐴𝑂𝐵), ℝ^, < ) → ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))))
51	47, 50	mtod 201	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ¬ 𝐴 < inf((𝐴𝑂𝐵), ℝ^*, < ))
52	9, 10, 51	xrnltled 11273	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝐴)
53	4	simp2d 1161	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝐴𝑅𝑤)
54	10	adantr 485	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝐴 ∈ ℝ^*)
55		ixxub.5	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴𝑅𝑤 → 𝐴 ≤ 𝑤))
56	54, 5, 55	syl2anc 595	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → (𝐴𝑅𝑤 → 𝐴 ≤ 𝑤))
57	53, 56	mpd 16	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝐴 ≤ 𝑤)
58	57	ralrimiva 3157	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ∀𝑤 ∈ (𝐴𝑂𝐵)𝐴 ≤ 𝑤)
59		infxrgelb 13357	. . . 4 ⊢ (((𝐴𝑂𝐵) ⊆ ℝ^* ∧ 𝐴 ∈ ℝ^) → (𝐴 ≤ inf((𝐴𝑂𝐵), ℝ^, < ) ↔ ∀𝑤 ∈ (𝐴𝑂𝐵)𝐴 ≤ 𝑤))
60	7, 10, 59	syl2anc 595	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴 ≤ inf((𝐴𝑂𝐵), ℝ^*, < ) ↔ ∀𝑤 ∈ (𝐴𝑂𝐵)𝐴 ≤ 𝑤))
61	58, 60	mpbird 260	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → 𝐴 ≤ inf((𝐴𝑂𝐵), ℝ^*, < ))
62	9, 10, 52, 61	xrletrid 13175	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^*, < ) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∃wex 1809 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 {crab 3416 ⊆ wss 3905 ∅c0 4286 class class class wbr 5109 (class class class)co 7410 ∈ cmpo 7412 infcinf 9397 ℝ^*cxr 11237 < clt 11238 ≤ cle 11239 ℚcq 12967

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9398 df-inf 9399 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-n0 12500 df-z 12587 df-uz 12858 df-q 12968

This theorem is referenced by: ioorf 25732 ioorinv2 25734 ioossioobi 46233

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