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Theorem xrsupexmnf 13290
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 4143 . . . . . 6 (𝑦 ∈ (𝐴 ∪ {-∞}) ↔ (𝑦𝐴𝑦 ∈ {-∞}))
2 simpr 484 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦𝐴 → ¬ 𝑥 < 𝑦))
3 velsn 4639 . . . . . . . . 9 (𝑦 ∈ {-∞} ↔ 𝑦 = -∞)
4 nltmnf 13115 . . . . . . . . . 10 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
5 breq2 5145 . . . . . . . . . . 11 (𝑦 = -∞ → (𝑥 < 𝑦𝑥 < -∞))
65notbid 318 . . . . . . . . . 10 (𝑦 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞))
74, 6syl5ibrcom 246 . . . . . . . . 9 (𝑥 ∈ ℝ* → (𝑦 = -∞ → ¬ 𝑥 < 𝑦))
83, 7biimtrid 241 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
98adantr 480 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
102, 9jaod 856 . . . . . 6 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → ((𝑦𝐴𝑦 ∈ {-∞}) → ¬ 𝑥 < 𝑦))
111, 10biimtrid 241 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦))
1211ex 412 . . . 4 (𝑥 ∈ ℝ* → ((𝑦𝐴 → ¬ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦)))
1312ralimdv2 3157 . . 3 (𝑥 ∈ ℝ* → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 → ∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦))
14 elun1 4171 . . . . . . 7 (𝑧𝐴𝑧 ∈ (𝐴 ∪ {-∞}))
1514anim1i 614 . . . . . 6 ((𝑧𝐴𝑦 < 𝑧) → (𝑧 ∈ (𝐴 ∪ {-∞}) ∧ 𝑦 < 𝑧))
1615reximi2 3073 . . . . 5 (∃𝑧𝐴 𝑦 < 𝑧 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)
1716imim2i 16 . . . 4 ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) → (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))
1817ralimi 3077 . . 3 (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))
1913, 18anim12d1 609 . 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 3075 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 844   = wceq 1533  wcel 2098  wral 3055  wrex 3064  cun 3941  {csn 4623   class class class wbr 5141  -∞cmnf 11250  *cxr 11251   < clt 11252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257
This theorem is referenced by:  xrsupss  13294
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