Theorem dfinfre 12221

Description: The infimum of a set of reals 𝐴. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)

Assertion

Ref	Expression
dfinfre	⊢ (𝐴 ⊆ ℝ → inf(𝐴, ℝ, < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))})

Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dfinfre

Step	Hyp	Ref	Expression
1		df-inf 9414	. 2 ⊢ inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, ◡ < )
2		df-sup 9413	. . 3 ⊢ sup(𝐴, ℝ, ◡ < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧))}
3		ssel2 3926	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
4		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
5		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5862	. . . . . . . . . . . 12 ⊢ (𝑥◡ < 𝑦 ↔ 𝑦 < 𝑥)
7	6	notbii 323	. . . . . . . . . . 11 ⊢ (¬ 𝑥◡ < 𝑦 ↔ ¬ 𝑦 < 𝑥)
8		lenlt 11313	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑥))
9	7, 8	bitr4id 293	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
10	3, 9	sylan2 605	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
11	10	ancoms 464	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
12	11	an32s 665	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
13	12	ralbidva 3183	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
14	5, 4	brcnv 5862	. . . . . . . . 9 ⊢ (𝑦◡ < 𝑥 ↔ 𝑥 < 𝑦)
15		vex 3454	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
16	5, 15	brcnv 5862	. . . . . . . . . 10 ⊢ (𝑦◡ < 𝑧 ↔ 𝑧 < 𝑦)
17	16	rexbii 3109	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧 ↔ ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)
18	14, 17	imbi12i 353	. . . . . . . 8 ⊢ ((𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧) ↔ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
19	18	ralbii 3108	. . . . . . 7 ⊢ (∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
20	19	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
21	13, 20	anbi12d 644	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → ((∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧)) ↔ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
22	21	rabbidva 3418	. . . 4 ⊢ (𝐴 ⊆ ℝ → {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧))} = {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))})
23	22	unieqd 4880	. . 3 ⊢ (𝐴 ⊆ ℝ → ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧))} = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))})
24	2, 23	eqtrid 2807	. 2 ⊢ (𝐴 ⊆ ℝ → sup(𝐴, ℝ, ◡ < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))})
25	1, 24	eqtrid 2807	1 ⊢ (𝐴 ⊆ ℝ → inf(𝐴, ℝ, < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3899  ∪ cuni 4867   class class class wbr 5103  ^◡ccnv 5654  supcsup 9411  infcinf 9412  ℝcr 11124   < clt 11268   ≤ cle 11269

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-sup 9413  df-inf 9414  df-xr 11272  df-le 11274

This theorem is used by: (None)