MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xrinfmexpnf Structured version   Visualization version   GIF version

Theorem xrinfmexpnf 12782
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 4039 . . . . . 6 (𝑦 ∈ (𝐴 ∪ {+∞}) ↔ (𝑦𝐴𝑦 ∈ {+∞}))
2 simpr 488 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦𝐴 → ¬ 𝑦 < 𝑥))
3 velsn 4532 . . . . . . . . 9 (𝑦 ∈ {+∞} ↔ 𝑦 = +∞)
4 pnfnlt 12606 . . . . . . . . . 10 (𝑥 ∈ ℝ* → ¬ +∞ < 𝑥)
5 breq1 5033 . . . . . . . . . . 11 (𝑦 = +∞ → (𝑦 < 𝑥 ↔ +∞ < 𝑥))
65notbid 321 . . . . . . . . . 10 (𝑦 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ +∞ < 𝑥))
74, 6syl5ibrcom 250 . . . . . . . . 9 (𝑥 ∈ ℝ* → (𝑦 = +∞ → ¬ 𝑦 < 𝑥))
83, 7syl5bi 245 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
98adantr 484 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
102, 9jaod 858 . . . . . 6 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → ((𝑦𝐴𝑦 ∈ {+∞}) → ¬ 𝑦 < 𝑥))
111, 10syl5bi 245 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥))
1211ex 416 . . . 4 (𝑥 ∈ ℝ* → ((𝑦𝐴 → ¬ 𝑦 < 𝑥) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥)))
1312ralimdv2 3090 . . 3 (𝑥 ∈ ℝ* → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 → ∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥))
14 elun1 4066 . . . . . . 7 (𝑧𝐴𝑧 ∈ (𝐴 ∪ {+∞}))
1514anim1i 618 . . . . . 6 ((𝑧𝐴𝑧 < 𝑦) → (𝑧 ∈ (𝐴 ∪ {+∞}) ∧ 𝑧 < 𝑦))
1615reximi2 3158 . . . . 5 (∃𝑧𝐴 𝑧 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)
1716imim2i 16 . . . 4 ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))
1817ralimi 3075 . . 3 (∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))
1913, 18anim12d1 613 . 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 3156 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 399  wo 846   = wceq 1542  wcel 2114  wral 3053  wrex 3054  cun 3841  {csn 4516   class class class wbr 5030  +∞cpnf 10750  *cxr 10752   < clt 10753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-xp 5531  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758
This theorem is referenced by:  xrinfmss  12786
  Copyright terms: Public domain W3C validator