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Theorem dfinfre 12144
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 9387 . 2 inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, < )
2 df-sup 9386 . . 3 sup(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))}
3 ssel2 3943 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → 𝑦 ∈ ℝ)
4 vex 3451 . . . . . . . . . . . . 13 𝑥 ∈ V
5 vex 3451 . . . . . . . . . . . . 13 𝑦 ∈ V
64, 5brcnv 5842 . . . . . . . . . . . 12 (𝑥 < 𝑦𝑦 < 𝑥)
76notbii 320 . . . . . . . . . . 11 𝑥 < 𝑦 ↔ ¬ 𝑦 < 𝑥)
8 lenlt 11241 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥𝑦 ↔ ¬ 𝑦 < 𝑥))
97, 8bitr4id 290 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (¬ 𝑥 < 𝑦𝑥𝑦))
103, 9sylan2 594 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑦𝐴)) → (¬ 𝑥 < 𝑦𝑥𝑦))
1110ancoms 460 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝑦𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 < 𝑦𝑥𝑦))
1211an32s 651 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦𝐴) → (¬ 𝑥 < 𝑦𝑥𝑦))
1312ralbidva 3169 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦𝐴 𝑥𝑦))
145, 4brcnv 5842 . . . . . . . . 9 (𝑦 < 𝑥𝑥 < 𝑦)
15 vex 3451 . . . . . . . . . . 11 𝑧 ∈ V
165, 15brcnv 5842 . . . . . . . . . 10 (𝑦 < 𝑧𝑧 < 𝑦)
1716rexbii 3094 . . . . . . . . 9 (∃𝑧𝐴 𝑦 < 𝑧 ↔ ∃𝑧𝐴 𝑧 < 𝑦)
1814, 17imbi12i 351 . . . . . . . 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 632 . . . . 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 3413 . . . 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 4883 . . 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 2785 . 2 (𝐴 ⊆ ℝ → sup(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))})
251, 24eqtrid 2785 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 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  {crab 3406  wss 3914   cuni 4869   class class class wbr 5109  ccnv 5636  supcsup 9384  infcinf 9385  cr 11058   < clt 11197  cle 11198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-sup 9386  df-inf 9387  df-xr 11201  df-le 11203
This theorem is referenced by: (None)
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