Theorem dfinfre 12103

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 9327	. 2 ⊢ inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, ◡ < )
2		df-sup 9326	. . 3 ⊢ sup(𝐴, ℝ, ◡ < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧))}
3		ssel2 3924	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
4		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
5		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5821	. . . . . . . . . . . 12 ⊢ (𝑥◡ < 𝑦 ↔ 𝑦 < 𝑥)
7	6	notbii 320	. . . . . . . . . . 11 ⊢ (¬ 𝑥◡ < 𝑦 ↔ ¬ 𝑦 < 𝑥)
8		lenlt 11191	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑥))
9	7, 8	bitr4id 290	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
10	3, 9	sylan2 593	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
11	10	ancoms 458	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
12	11	an32s 652	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
13	12	ralbidva 3153	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
14	5, 4	brcnv 5821	. . . . . . . . 9 ⊢ (𝑦◡ < 𝑥 ↔ 𝑥 < 𝑦)
15		vex 3440	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
16	5, 15	brcnv 5821	. . . . . . . . . 10 ⊢ (𝑦◡ < 𝑧 ↔ 𝑧 < 𝑦)
17	16	rexbii 3079	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧 ↔ ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)
18	14, 17	imbi12i 350	. . . . . . . 8 ⊢ ((𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧) ↔ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
19	18	ralbii 3078	. . . . . . 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 632	. . . . 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 3401	. . . 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 4869	. . 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 2778	. 2 ⊢ (𝐴 ⊆ ℝ → sup(𝐴, ℝ, ◡ < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))})
25	1, 24	eqtrid 2778	1 ⊢ (𝐴 ⊆ ℝ → inf(𝐴, ℝ, < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  {crab 3395   ⊆ wss 3897  ∪ cuni 4856   class class class wbr 5089  ^◡ccnv 5613  supcsup 9324  infcinf 9325  ℝcr 11005   < clt 11146   ≤ cle 11147

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-sup 9326  df-inf 9327  df-xr 11150  df-le 11152

This theorem is referenced by: (None)