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Theorem dfinfre 11216
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 8516 . 2 inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, < )
2 df-sup 8515 . . 3 sup(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))}
3 ssel2 3739 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑦𝐴) → 𝑦 ∈ ℝ)
4 lenlt 10328 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥𝑦 ↔ ¬ 𝑦 < 𝑥))
5 vex 3343 . . . . . . . . . . . . 13 𝑥 ∈ V
6 vex 3343 . . . . . . . . . . . . 13 𝑦 ∈ V
75, 6brcnv 5460 . . . . . . . . . . . 12 (𝑥 < 𝑦𝑦 < 𝑥)
87notbii 309 . . . . . . . . . . 11 𝑥 < 𝑦 ↔ ¬ 𝑦 < 𝑥)
94, 8syl6rbbr 279 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (¬ 𝑥 < 𝑦𝑥𝑦))
103, 9sylan2 492 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑦𝐴)) → (¬ 𝑥 < 𝑦𝑥𝑦))
1110ancoms 468 . . . . . . . 8 (((𝐴 ⊆ ℝ ∧ 𝑦𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 < 𝑦𝑥𝑦))
1211an32s 881 . . . . . . 7 (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦𝐴) → (¬ 𝑥 < 𝑦𝑥𝑦))
1312ralbidva 3123 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦𝐴 𝑥𝑦))
146, 5brcnv 5460 . . . . . . . . 9 (𝑦 < 𝑥𝑥 < 𝑦)
15 vex 3343 . . . . . . . . . . 11 𝑧 ∈ V
166, 15brcnv 5460 . . . . . . . . . 10 (𝑦 < 𝑧𝑧 < 𝑦)
1716rexbii 3179 . . . . . . . . 9 (∃𝑧𝐴 𝑦 < 𝑧 ↔ ∃𝑧𝐴 𝑧 < 𝑦)
1814, 17imbi12i 339 . . . . . . . 8 ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
1918ralbii 3118 . . . . . . 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 749 . . . . 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 3328 . . . 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 4598 . . 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 2806 . 2 (𝐴 ⊆ ℝ → sup(𝐴, ℝ, < ) = {𝑥 ∈ ℝ ∣ (∀𝑦𝐴 𝑥𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))})
251, 24syl5eq 2806 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 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  wss 3715   cuni 4588   class class class wbr 4804  ccnv 5265  supcsup 8513  infcinf 8514  cr 10147   < clt 10286  cle 10287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-sup 8515  df-inf 8516  df-xr 10290  df-le 10292
This theorem is referenced by: (None)
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