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Theorem ixxlb 13194

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 13181	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵)))
3	2	3adant3 1131	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵)))
4	3	biimpa 477	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵))
5	4	simp1d 1141	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝑤 ∈ ℝ^*)
6	5	ex 413	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ ℝ^*))
7	6	ssrdv 3937	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴𝑂𝐵) ⊆ ℝ^*)
8		infxrcl 13160	. . 3 ⊢ ((𝐴𝑂𝐵) ⊆ ℝ^* → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^)
9	7, 8	syl 17	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^)
10		simp1 1135	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → 𝐴 ∈ ℝ^*)
11		simprr 770	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))
12	7	ad2antrr 723	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → (𝐴𝑂𝐵) ⊆ ℝ^)
13		qre 12786	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ℚ → 𝑤 ∈ ℝ)
14	13	rexrd 11118	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℚ → 𝑤 ∈ ℝ^*)
15	14	ad2antlr 724	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → 𝑤 ∈ ℝ^)
16		simprl 768	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝐴 < 𝑤)
17	10	ad2antrr 723	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → (𝐴 < 𝑤 → 𝐴𝑅𝑤))
20	16, 19	mpd 15	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝐴𝑅𝑤)
21	9	ad2antrr 723	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^*)
22		simpll2 1212	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → 𝐵 ∈ ℝ^)
23		simp3 1137	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴𝑂𝐵) ≠ ∅)
24		n0 4292	. . . . . . . . . . . . . 14 ⊢ ((𝐴𝑂𝐵) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝐴𝑂𝐵))
25	23, 24	sylib 217	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ∃𝑤 𝑤 ∈ (𝐴𝑂𝐵))
26	9	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^)
27		simpl2 1191	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝐵 ∈ ℝ^*)
28		infxrlb 13161	. . . . . . . . . . . . . . 15 ⊢ (((𝐴𝑂𝐵) ⊆ ℝ^* ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝑤)
29	7, 28	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝑤)
30	4	simp3d 1143	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → (𝑤𝑆𝐵 → 𝑤 ≤ 𝐵))
33	30, 32	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝑤 ≤ 𝐵)
34	26, 5, 27, 29, 33	xrletrd 12989	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝐵)
35	25, 34	exlimddv 1937	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝐵)
36	35	ad2antrr 723	. . . . . . . . . . 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 12988	. . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → (𝑤 < 𝐵 → 𝑤𝑆𝐵))
40	37, 39	mpd 15	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝑤𝑆𝐵)
41	3	ad2antrr 723	. . . . . . . . 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 1341	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝑤 ∈ (𝐴𝑂𝐵))
43	12, 42, 28	syl2anc 584	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → inf((𝐴𝑂𝐵), ℝ^, < ) ≤ 𝑤)
44	21, 15	xrlenltd 11134	. . . . . . 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 231	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → ¬ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))
46	11, 45	pm2.65da 814	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) → ¬ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < )))
47	46	nrexdv 3142	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ¬ ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < )))
48		qbtwnxr 13027	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^ ∧ 𝐴 < inf((𝐴𝑂𝐵), ℝ^, < )) → ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < )))
49	48	3expia 1120	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^) → (𝐴 < inf((𝐴𝑂𝐵), ℝ^, < ) → ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))))
50	10, 9, 49	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴 < inf((𝐴𝑂𝐵), ℝ^, < ) → ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))))
51	47, 50	mtod 197	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ¬ 𝐴 < inf((𝐴𝑂𝐵), ℝ^*, < ))
52	9, 10, 51	xrnltled 11136	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝐴)
53	4	simp2d 1142	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝐴𝑅𝑤)
54	10	adantr 481	. . . . . 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 584	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → (𝐴𝑅𝑤 → 𝐴 ≤ 𝑤))
57	53, 56	mpd 15	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝐴 ≤ 𝑤)
58	57	ralrimiva 3139	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ∀𝑤 ∈ (𝐴𝑂𝐵)𝐴 ≤ 𝑤)
59		infxrgelb 13162	. . . 4 ⊢ (((𝐴𝑂𝐵) ⊆ ℝ^* ∧ 𝐴 ∈ ℝ^) → (𝐴 ≤ inf((𝐴𝑂𝐵), ℝ^, < ) ↔ ∀𝑤 ∈ (𝐴𝑂𝐵)𝐴 ≤ 𝑤))
60	7, 10, 59	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴 ≤ inf((𝐴𝑂𝐵), ℝ^*, < ) ↔ ∀𝑤 ∈ (𝐴𝑂𝐵)𝐴 ≤ 𝑤))
61	58, 60	mpbird 256	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → 𝐴 ≤ inf((𝐴𝑂𝐵), ℝ^*, < ))
62	9, 10, 52, 61	xrletrid 12982	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^*, < ) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3403 ⊆ wss 3897 ∅c0 4268 class class class wbr 5089 (class class class)co 7329 ∈ cmpo 7331 infcinf 9290 ℝ^*cxr 11101 < clt 11102 ≤ cle 11103 ℚcq 12781

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-1st 7891 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-er 8561 df-en 8797 df-dom 8798 df-sdom 8799 df-sup 9291 df-inf 9292 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-n0 12327 df-z 12413 df-uz 12676 df-q 12782

This theorem is referenced by: ioorf 24835 ioorinv2 24837 ioossioobi 43380

W3C validator