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Theorem xrsupexmnf 12684
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 4109 . . . . . 6 (𝑦 ∈ (𝐴 ∪ {-∞}) ↔ (𝑦𝐴𝑦 ∈ {-∞}))
2 simpr 488 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦𝐴 → ¬ 𝑥 < 𝑦))
3 velsn 4564 . . . . . . . . 9 (𝑦 ∈ {-∞} ↔ 𝑦 = -∞)
4 nltmnf 12510 . . . . . . . . . 10 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
5 breq2 5051 . . . . . . . . . . 11 (𝑦 = -∞ → (𝑥 < 𝑦𝑥 < -∞))
65notbid 321 . . . . . . . . . 10 (𝑦 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞))
74, 6syl5ibrcom 250 . . . . . . . . 9 (𝑥 ∈ ℝ* → (𝑦 = -∞ → ¬ 𝑥 < 𝑦))
83, 7syl5bi 245 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
98adantr 484 . . . . . . 7 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦))
102, 9jaod 856 . . . . . 6 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → ((𝑦𝐴𝑦 ∈ {-∞}) → ¬ 𝑥 < 𝑦))
111, 10syl5bi 245 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝑦𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦))
1211ex 416 . . . 4 (𝑥 ∈ ℝ* → ((𝑦𝐴 → ¬ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦)))
1312ralimdv2 3170 . . 3 (𝑥 ∈ ℝ* → (∀𝑦𝐴 ¬ 𝑥 < 𝑦 → ∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦))
14 elun1 4136 . . . . . . 7 (𝑧𝐴𝑧 ∈ (𝐴 ∪ {-∞}))
1514anim1i 617 . . . . . 6 ((𝑧𝐴𝑦 < 𝑧) → (𝑧 ∈ (𝐴 ∪ {-∞}) ∧ 𝑦 < 𝑧))
1615reximi2 3238 . . . . 5 (∃𝑧𝐴 𝑦 < 𝑧 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)
1716imim2i 16 . . . 4 ((𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) → (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))
1817ralimi 3154 . . 3 (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))
1913, 18anim12d1 612 . 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 3236 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 399  wo 844   = wceq 1538  wcel 2115  wral 3132  wrex 3133  cun 3916  {csn 4548   class class class wbr 5047  -∞cmnf 10658  *cxr 10659   < clt 10660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-resscn 10579
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665
This theorem is referenced by:  xrsupss  12688
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