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Mirrors > Home > MPE Home > Th. List > mnflt | Structured version Visualization version GIF version |
Description: Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
mnflt | ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ -∞ = -∞ | |
2 | olc 864 | . . . 4 ⊢ ((-∞ = -∞ ∧ 𝐴 ∈ ℝ) → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℝ → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))) |
4 | 3 | olcd 870 | . 2 ⊢ (𝐴 ∈ ℝ → ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))) |
5 | mnfxr 10698 | . . 3 ⊢ -∞ ∈ ℝ* | |
6 | rexr 10687 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
7 | ltxr 12511 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))) | |
8 | 5, 6, 7 | sylancr 589 | . 2 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))) |
9 | 4, 8 | mpbird 259 | 1 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ℝcr 10536 <ℝ cltrr 10541 +∞cpnf 10672 -∞cmnf 10673 ℝ*cxr 10674 < clt 10675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 |
This theorem is referenced by: mnfltd 12520 mnflt0 12521 mnfltxr 12523 xrlttri 12533 xrlttr 12534 xrrebnd 12562 xrre3 12565 qbtwnxr 12594 xrsupsslem 12701 xrub 12706 elico2 12801 elicc2 12802 ioomax 12812 elioomnf 12833 difreicc 12871 icopnfcld 23376 iocmnfcld 23377 xrtgioo 23414 bndth 23562 mbfmax 24250 itg2seq 24343 ellogdm 25222 esumcvgsum 31347 dya2iocbrsiga 31533 dya2icobrsiga 31534 orvclteel 31730 iooelexlt 34646 itg2addnclem 34958 asindmre 34992 dvasin 34993 dvacos 34994 rfcnpre4 41311 infrpge 41639 infxr 41655 infxrunb2 41656 infleinflem2 41659 icccncfext 42190 fourierdlem113 42524 fouriersw 42536 iccpartigtl 43603 |
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