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Theorem xrinfmexpnf 12687
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 4122 . . . . . 6 (𝑦 ∈ (𝐴 ∪ {+∞}) ↔ (𝑦𝐴𝑦 ∈ {+∞}))
2 simpr 485 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦𝐴 → ¬ 𝑦 < 𝑥))
3 velsn 4573 . . . . . . . . 9 (𝑦 ∈ {+∞} ↔ 𝑦 = +∞)
4 pnfnlt 12511 . . . . . . . . . 10 (𝑥 ∈ ℝ* → ¬ +∞ < 𝑥)
5 breq1 5060 . . . . . . . . . . 11 (𝑦 = +∞ → (𝑦 < 𝑥 ↔ +∞ < 𝑥))
65notbid 319 . . . . . . . . . 10 (𝑦 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ +∞ < 𝑥))
74, 6syl5ibrcom 248 . . . . . . . . 9 (𝑥 ∈ ℝ* → (𝑦 = +∞ → ¬ 𝑦 < 𝑥))
83, 7syl5bi 243 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
98adantr 481 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ {+∞} → ¬ 𝑦 < 𝑥))
102, 9jaod 853 . . . . . 6 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → ((𝑦𝐴𝑦 ∈ {+∞}) → ¬ 𝑦 < 𝑥))
111, 10syl5bi 243 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑦 < 𝑥)) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥))
1211ex 413 . . . 4 (𝑥 ∈ ℝ* → ((𝑦𝐴 → ¬ 𝑦 < 𝑥) → (𝑦 ∈ (𝐴 ∪ {+∞}) → ¬ 𝑦 < 𝑥)))
1312ralimdv2 3173 . . 3 (𝑥 ∈ ℝ* → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 → ∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥))
14 elun1 4149 . . . . . . 7 (𝑧𝐴𝑧 ∈ (𝐴 ∪ {+∞}))
1514anim1i 614 . . . . . 6 ((𝑧𝐴𝑧 < 𝑦) → (𝑧 ∈ (𝐴 ∪ {+∞}) ∧ 𝑧 < 𝑦))
1615reximi2 3241 . . . . 5 (∃𝑧𝐴 𝑧 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)
1716imim2i 16 . . . 4 ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))
1817ralimi 3157 . . 3 (∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦))
1913, 18anim12d1 609 . 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 3239 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 396  wo 841   = wceq 1528  wcel 2105  wral 3135  wrex 3136  cun 3931  {csn 4557   class class class wbr 5057  +∞cpnf 10660  *cxr 10662   < clt 10663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668
This theorem is referenced by:  xrinfmss  12691
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