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Theorem xrinf0 12206
Description: The infimum of the empty set under the extended reals is positive infinity. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 5-Sep-2020.)
Assertion
Ref Expression
xrinf0 inf(∅, ℝ*, < ) = +∞

Proof of Theorem xrinf0
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrltso 12012 . . . 4 < Or ℝ*
21a1i 11 . . 3 (⊤ → < Or ℝ*)
3 pnfxr 10130 . . . 4 +∞ ∈ ℝ*
43a1i 11 . . 3 (⊤ → +∞ ∈ ℝ*)
5 noel 3952 . . . . 5 ¬ 𝑦 ∈ ∅
65pm2.21i 116 . . . 4 (𝑦 ∈ ∅ → ¬ 𝑦 < +∞)
76adantl 481 . . 3 ((⊤ ∧ 𝑦 ∈ ∅) → ¬ 𝑦 < +∞)
8 pnfnlt 12000 . . . . . 6 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
98pm2.21d 118 . . . . 5 (𝑦 ∈ ℝ* → (+∞ < 𝑦 → ∃𝑧 ∈ ∅ 𝑧 < 𝑦))
109imp 444 . . . 4 ((𝑦 ∈ ℝ* ∧ +∞ < 𝑦) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)
1110adantl 481 . . 3 ((⊤ ∧ (𝑦 ∈ ℝ* ∧ +∞ < 𝑦)) → ∃𝑧 ∈ ∅ 𝑧 < 𝑦)
122, 4, 7, 11eqinfd 8432 . 2 (⊤ → inf(∅, ℝ*, < ) = +∞)
1312trud 1533 1 inf(∅, ℝ*, < ) = +∞
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1523  wtru 1524  wcel 2030  wrex 2942  c0 3948   class class class wbr 4685   Or wor 5063  infcinf 8388  +∞cpnf 10109  *cxr 10111   < clt 10112
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-8 2032  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-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  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-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117
This theorem is referenced by:  ramcl2lem  15760  infleinf  39901  infxrpnf  39987  supxrltinfxr  39990  supminfxr  40007
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