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Theorem xrinfmexpnf 13345
Description: Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)
Assertion
Ref Expression
xrinfmexpnf (∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem xrinfmexpnf
StepHypRef Expression
1 elun 4163 . . . . . 6 (𝑦 ∈ (𝐴 ∪ {+∞}) ↔ (𝑦𝐴𝑦 ∈ {+∞}))
2 simpr 484 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦𝐴 → ¬ 𝑦 < 𝑥))
3 velsn 4647 . . . . . . . . 9 (𝑦 ∈ {+∞} ↔ 𝑦 = +∞)
4 pnfnlt 13168 . . . . . . . . . 10 (𝑥 ∈ ℝ* → ¬ +∞ < 𝑥)
5 breq1 5151 . . . . . . . . . . 11 (𝑦 = +∞ → (𝑦 < 𝑥 ↔ +∞ < 𝑥))
65notbid 318 . . . . . . . . . 10 (𝑦 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ +∞ < 𝑥))
74, 6syl5ibrcom 247 . . . . . . . . 9 (𝑥 ∈ ℝ* → (𝑦 = +∞ → ¬ 𝑦 < 𝑥))
83, 7biimtrid 242 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
98adantr 480 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
102, 9jaod 859 . . . . . 6 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → ((𝑦𝐴𝑦 ∈ {+∞}) → ¬ 𝑦 < 𝑥))
111, 10biimtrid 242 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥))
1211ex 412 . . . 4 (𝑥 ∈ ℝ* → ((𝑦𝐴 → ¬ 𝑦 < 𝑥) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥)))
1312ralimdv2 3161 . . 3 (𝑥 ∈ ℝ* → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 → ∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥))
14 elun1 4192 . . . . . . 7 (𝑧𝐴𝑧 ∈ (𝐴 ∪ {+∞}))
1514anim1i 615 . . . . . 6 ((𝑧𝐴𝑧 < 𝑦) → (𝑧 ∈ (𝐴 ∪ {+∞}) ∧ 𝑧 < 𝑦))
1615reximi2 3077 . . . . 5 (∃𝑧𝐴 𝑧 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)
1716imim2i 16 . . . 4 ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))
1817ralimi 3081 . . 3 (∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))
1913, 18anim12d1 610 . 2 (𝑥 ∈ ℝ* → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → (∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))))
2019reximia 3079 1 (∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cun 3961  {csn 4631   class class class wbr 5148  +∞cpnf 11290  *cxr 11292   < clt 11293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298
This theorem is referenced by:  xrinfmss  13349
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