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Theorem dfinfre 11042
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 8390 . 2 inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, < )
2 df-sup 8389 . . 3 sup(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))}
3 ssel2 3631 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → 𝑦 ∈ ℝ)
4 lenlt 10154 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥𝑦 ↔ ¬ 𝑦 < 𝑥))
5 vex 3234 . . . . . . . . . . . . 13 𝑥 ∈ V
6 vex 3234 . . . . . . . . . . . . 13 𝑦 ∈ V
75, 6brcnv 5337 . . . . . . . . . . . 12 (𝑥 < 𝑦𝑦 < 𝑥)
87notbii 309 . . . . . . . . . . 11 𝑥 < 𝑦 ↔ ¬ 𝑦 < 𝑥)
94, 8syl6rbbr 279 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (¬ 𝑥 < 𝑦𝑥𝑦))
103, 9sylan2 490 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑦𝐴)) → (¬ 𝑥 < 𝑦𝑥𝑦))
1110ancoms 468 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝑦𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 < 𝑦𝑥𝑦))
1211an32s 863 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦𝐴) → (¬ 𝑥 < 𝑦𝑥𝑦))
1312ralbidva 3014 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦𝐴 𝑥𝑦))
146, 5brcnv 5337 . . . . . . . . 9 (𝑦 < 𝑥𝑥 < 𝑦)
15 vex 3234 . . . . . . . . . . 11 𝑧 ∈ V
166, 15brcnv 5337 . . . . . . . . . 10 (𝑦 < 𝑧𝑧 < 𝑦)
1716rexbii 3070 . . . . . . . . 9 (∃𝑧𝐴 𝑦 < 𝑧 ↔ ∃𝑧𝐴 𝑧 < 𝑦)
1814, 17imbi12i 339 . . . . . . . 8 ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
1918ralbii 3009 . . . . . . 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 747 . . . . 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 3219 . . . 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 4478 . . 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 2697 . 2 (𝐴 ⊆ ℝ → sup(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))})
251, 24syl5eq 2697 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  wss 3607   cuni 4468   class class class wbr 4685  ccnv 5142  supcsup 8387  infcinf 8388  cr 9973   < clt 10112  cle 10113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-sup 8389  df-inf 8390  df-xr 10116  df-le 10118
This theorem is referenced by: (None)
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