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Theorem dfinfre 11258
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 8556 . 2 inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, < )
2 df-sup 8555 . . 3 sup(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))}
3 ssel2 3756 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → 𝑦 ∈ ℝ)
4 lenlt 10370 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥𝑦 ↔ ¬ 𝑦 < 𝑥))
5 vex 3353 . . . . . . . . . . . . 13 𝑥 ∈ V
6 vex 3353 . . . . . . . . . . . . 13 𝑦 ∈ V
75, 6brcnv 5473 . . . . . . . . . . . 12 (𝑥 < 𝑦𝑦 < 𝑥)
87notbii 311 . . . . . . . . . . 11 𝑥 < 𝑦 ↔ ¬ 𝑦 < 𝑥)
94, 8syl6rbbr 281 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (¬ 𝑥 < 𝑦𝑥𝑦))
103, 9sylan2 586 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑦𝐴)) → (¬ 𝑥 < 𝑦𝑥𝑦))
1110ancoms 450 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝑦𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 < 𝑦𝑥𝑦))
1211an32s 642 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦𝐴) → (¬ 𝑥 < 𝑦𝑥𝑦))
1312ralbidva 3132 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦𝐴 𝑥𝑦))
146, 5brcnv 5473 . . . . . . . . 9 (𝑦 < 𝑥𝑥 < 𝑦)
15 vex 3353 . . . . . . . . . . 11 𝑧 ∈ V
166, 15brcnv 5473 . . . . . . . . . 10 (𝑦 < 𝑧𝑧 < 𝑦)
1716rexbii 3188 . . . . . . . . 9 (∃𝑧𝐴 𝑦 < 𝑧 ↔ ∃𝑧𝐴 𝑧 < 𝑦)
1814, 17imbi12i 341 . . . . . . . 8 ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
1918ralbii 3127 . . . . . . 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 624 . . . . 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 3337 . . . 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 4604 . . 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, 23syl5eq 2811 . 2 (𝐴 ⊆ ℝ → sup(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))})
251, 24syl5eq 2811 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  {crab 3059  wss 3732   cuni 4594   class class class wbr 4809  ccnv 5276  supcsup 8553  infcinf 8554  cr 10188   < clt 10328  cle 10329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-xp 5283  df-cnv 5285  df-sup 8555  df-inf 8556  df-xr 10332  df-le 10334
This theorem is referenced by: (None)
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