Theorem dfinfre 9029

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 7087	. 2 ⊢ inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, ◡ < )
2		df-sup 7086	. . 3 ⊢ sup(𝐴, ℝ, ◡ < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧))}
3		ssel2 3188	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
4		vex 2775	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
5		vex 2775	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 4861	. . . . . . . . . . . 12 ⊢ (𝑥◡ < 𝑦 ↔ 𝑦 < 𝑥)
7	6	notbii 670	. . . . . . . . . . 11 ⊢ (¬ 𝑥◡ < 𝑦 ↔ ¬ 𝑦 < 𝑥)
8		lenlt 8148	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 ≤ 𝑦 ↔ ¬ 𝑦 < 𝑥))
9	7, 8	bitr4id 199	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
10	3, 9	sylan2 286	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴)) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
11	10	ancoms 268	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
12	11	an32s 568	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥◡ < 𝑦 ↔ 𝑥 ≤ 𝑦))
13	12	ralbidva 2502	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
14	5, 4	brcnv 4861	. . . . . . . . 9 ⊢ (𝑦◡ < 𝑥 ↔ 𝑥 < 𝑦)
15		vex 2775	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
16	5, 15	brcnv 4861	. . . . . . . . . 10 ⊢ (𝑦◡ < 𝑧 ↔ 𝑧 < 𝑦)
17	16	rexbii 2513	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧 ↔ ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)
18	14, 17	imbi12i 239	. . . . . . . 8 ⊢ ((𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧) ↔ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
19	18	ralbii 2512	. . . . . . 7 ⊢ (∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
20	19	a1i 9	. . . . . 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 473	. . . . 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 2760	. . . 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 3861	. . 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 2250	. 2 ⊢ (𝐴 ⊆ ℝ → sup(𝐴, ℝ, ◡ < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))})
25	1, 24	eqtrid 2250	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   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2176  ∀wral 2484  ∃wrex 2485  {crab 2488   ⊆ wss 3166  ∪ cuni 3850   class class class wbr 4044  ^◡ccnv 4674  supcsup 7084  infcinf 7085  ℝcr 7924   < clt 8107   ≤ cle 8108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-sup 7086  df-inf 7087  df-xr 8111  df-le 8113

This theorem is referenced by: (None)