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Theorem dfinfre 12191
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 ∣ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))})
Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dfinfre
StepHypRef Expression
1 df-inf 9434 . 2 inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, < )
2 df-sup 9433 . . 3 sup(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))}
3 ssel2 3976 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → 𝑦 ∈ ℝ)
4 vex 3478 . . . . . . . . . . . . 13 𝑥 ∈ V
5 vex 3478 . . . . . . . . . . . . 13 𝑦 ∈ V
64, 5brcnv 5880 . . . . . . . . . . . 12 (𝑥 < 𝑦𝑦 < 𝑥)
76notbii 319 . . . . . . . . . . 11 𝑥 < 𝑦 ↔ ¬ 𝑦 < 𝑥)
8 lenlt 11288 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥𝑦 ↔ ¬ 𝑦 < 𝑥))
97, 8bitr4id 289 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (¬ 𝑥 < 𝑦𝑥𝑦))
103, 9sylan2 593 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑦𝐴)) → (¬ 𝑥 < 𝑦𝑥𝑦))
1110ancoms 459 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝑦𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 < 𝑦𝑥𝑦))
1211an32s 650 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦𝐴) → (¬ 𝑥 < 𝑦𝑥𝑦))
1312ralbidva 3175 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦𝐴 𝑥𝑦))
145, 4brcnv 5880 . . . . . . . . 9 (𝑦 < 𝑥𝑥 < 𝑦)
15 vex 3478 . . . . . . . . . . 11 𝑧 ∈ V
165, 15brcnv 5880 . . . . . . . . . 10 (𝑦 < 𝑧𝑧 < 𝑦)
1716rexbii 3094 . . . . . . . . 9 (∃𝑧𝐴 𝑦 < 𝑧 ↔ ∃𝑧𝐴 𝑧 < 𝑦)
1814, 17imbi12i 350 . . . . . . . 8 ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
1918ralbii 3093 . . . . . . 7 (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
2019a1i 11 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2113, 20anbi12d 631 . . . . 5 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → ((∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)) ↔ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
2221rabbidva 3439 . . . 4 (𝐴 ⊆ ℝ → {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))} = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))})
2322unieqd 4921 . . 3 (𝐴 ⊆ ℝ → {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))} = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))})
242, 23eqtrid 2784 . 2 (𝐴 ⊆ ℝ → sup(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))})
251, 24eqtrid 2784 1 (𝐴 ⊆ ℝ → inf(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  wrex 3070  {crab 3432  wss 3947   cuni 4907   class class class wbr 5147  ccnv 5674  supcsup 9431  infcinf 9432  cr 11105   < clt 11244  cle 11245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-sup 9433  df-inf 9434  df-xr 11248  df-le 11250
This theorem is referenced by: (None)
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