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Theorem xrsupexmnf 12173
Description: Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)
Assertion
Ref Expression
xrsupexmnf (∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)))
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem xrsupexmnf
StepHypRef Expression
1 elun 3786 . . . . . 6 (𝑦 ∈ (𝐴 ∪ {-∞}) ↔ (𝑦𝐴𝑦 ∈ {-∞}))
2 simpr 476 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦𝐴 → ¬ 𝑥 < 𝑦))
3 velsn 4226 . . . . . . . . 9 (𝑦 ∈ {-∞} ↔ 𝑦 = -∞)
4 nltmnf 12001 . . . . . . . . . 10 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
5 breq2 4689 . . . . . . . . . . 11 (𝑦 = -∞ → (𝑥 < 𝑦𝑥 < -∞))
65notbid 307 . . . . . . . . . 10 (𝑦 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞))
74, 6syl5ibrcom 237 . . . . . . . . 9 (𝑥 ∈ ℝ* → (𝑦 = -∞ → ¬ 𝑥 < 𝑦))
83, 7syl5bi 232 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
98adantr 480 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
102, 9jaod 394 . . . . . 6 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → ((𝑦𝐴𝑦 ∈ {-∞}) → ¬ 𝑥 < 𝑦))
111, 10syl5bi 232 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦))
1211ex 449 . . . 4 (𝑥 ∈ ℝ* → ((𝑦𝐴 → ¬ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦)))
1312ralimdv2 2990 . . 3 (𝑥 ∈ ℝ* → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 → ∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦))
14 elun1 3813 . . . . . . . 8 (𝑧𝐴𝑧 ∈ (𝐴 ∪ {-∞}))
1514anim1i 591 . . . . . . 7 ((𝑧𝐴𝑦 < 𝑧) → (𝑧 ∈ (𝐴 ∪ {-∞}) ∧ 𝑦 < 𝑧))
1615reximi2 3039 . . . . . 6 (∃𝑧𝐴 𝑦 < 𝑧 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)
1716imim2i 16 . . . . 5 ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) → (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))
1817ralimi 2981 . . . 4 (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))
1918a1i 11 . . 3 (𝑥 ∈ ℝ* → (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)))
2013, 19anim12d 585 . 2 (𝑥 ∈ ℝ* → ((∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)) → (∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))))
2120reximia 3038 1 (∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cun 3605  {csn 4210   class class class wbr 4685  -∞cmnf 10110  *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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-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-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-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117
This theorem is referenced by:  xrsupss  12177
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