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Theorem xrinfmexpnf 13285
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 4149 . . . . . 6 (𝑦 ∈ (𝐴 ∪ {+∞}) ↔ (𝑦𝐴𝑦 ∈ {+∞}))
2 simpr 486 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦𝐴 → ¬ 𝑦 < 𝑥))
3 velsn 4645 . . . . . . . . 9 (𝑦 ∈ {+∞} ↔ 𝑦 = +∞)
4 pnfnlt 13108 . . . . . . . . . 10 (𝑥 ∈ ℝ* → ¬ +∞ < 𝑥)
5 breq1 5152 . . . . . . . . . . 11 (𝑦 = +∞ → (𝑦 < 𝑥 ↔ +∞ < 𝑥))
65notbid 318 . . . . . . . . . 10 (𝑦 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ +∞ < 𝑥))
74, 6syl5ibrcom 246 . . . . . . . . 9 (𝑥 ∈ ℝ* → (𝑦 = +∞ → ¬ 𝑦 < 𝑥))
83, 7biimtrid 241 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
98adantr 482 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
102, 9jaod 858 . . . . . 6 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → ((𝑦𝐴𝑦 ∈ {+∞}) → ¬ 𝑦 < 𝑥))
111, 10biimtrid 241 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥))
1211ex 414 . . . 4 (𝑥 ∈ ℝ* → ((𝑦𝐴 → ¬ 𝑦 < 𝑥) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥)))
1312ralimdv2 3164 . . 3 (𝑥 ∈ ℝ* → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 → ∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥))
14 elun1 4177 . . . . . . 7 (𝑧𝐴𝑧 ∈ (𝐴 ∪ {+∞}))
1514anim1i 616 . . . . . 6 ((𝑧𝐴𝑧 < 𝑦) → (𝑧 ∈ (𝐴 ∪ {+∞}) ∧ 𝑧 < 𝑦))
1615reximi2 3080 . . . . 5 (∃𝑧𝐴 𝑧 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)
1716imim2i 16 . . . 4 ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))
1817ralimi 3084 . . 3 (∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))
1913, 18anim12d1 611 . 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 3082 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 397  wo 846   = wceq 1542  wcel 2107  wral 3062  wrex 3071  cun 3947  {csn 4629   class class class wbr 5149  +∞cpnf 11245  *cxr 11247   < clt 11248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253
This theorem is referenced by:  xrinfmss  13289
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