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Theorem xrsupexmnf 13344
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 4163 . . . . . 6 (𝑦 ∈ (𝐴 ∪ {-∞}) ↔ (𝑦𝐴𝑦 ∈ {-∞}))
2 simpr 484 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦𝐴 → ¬ 𝑥 < 𝑦))
3 velsn 4647 . . . . . . . . 9 (𝑦 ∈ {-∞} ↔ 𝑦 = -∞)
4 nltmnf 13169 . . . . . . . . . 10 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
5 breq2 5152 . . . . . . . . . . 11 (𝑦 = -∞ → (𝑥 < 𝑦𝑥 < -∞))
65notbid 318 . . . . . . . . . 10 (𝑦 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞))
74, 6syl5ibrcom 247 . . . . . . . . 9 (𝑥 ∈ ℝ* → (𝑦 = -∞ → ¬ 𝑥 < 𝑦))
83, 7biimtrid 242 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
98adantr 480 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
102, 9jaod 859 . . . . . 6 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → ((𝑦𝐴𝑦 ∈ {-∞}) → ¬ 𝑥 < 𝑦))
111, 10biimtrid 242 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦))
1211ex 412 . . . 4 (𝑥 ∈ ℝ* → ((𝑦𝐴 → ¬ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦)))
1312ralimdv2 3161 . . 3 (𝑥 ∈ ℝ* → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 → ∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦))
14 elun1 4192 . . . . . . 7 (𝑧𝐴𝑧 ∈ (𝐴 ∪ {-∞}))
1514anim1i 615 . . . . . 6 ((𝑧𝐴𝑦 < 𝑧) → (𝑧 ∈ (𝐴 ∪ {-∞}) ∧ 𝑦 < 𝑧))
1615reximi2 3077 . . . . 5 (∃𝑧𝐴 𝑦 < 𝑧 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)
1716imim2i 16 . . . 4 ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) → (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))
1817ralimi 3081 . . 3 (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))
1913, 18anim12d1 610 . 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))))
2019reximia 3079 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  wa 395  wo 847   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cun 3961  {csn 4631   class class class wbr 5148  -∞cmnf 11291  *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-10 2139  ax-11 2155  ax-12 2175  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-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  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-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298
This theorem is referenced by:  xrsupss  13348
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