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Mirrors > Home > ILE Home > Th. List > mnflt | 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 2177 | . . . 4 ⊢ -∞ = -∞ | |
2 | olc 711 | . . . 4 ⊢ ((-∞ = -∞ ∧ 𝐴 ∈ ℝ) → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))) | |
3 | 1, 2 | mpan 424 | . . 3 ⊢ (𝐴 ∈ ℝ → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))) |
4 | 3 | olcd 734 | . 2 ⊢ (𝐴 ∈ ℝ → ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))) |
5 | mnfxr 8013 | . . 3 ⊢ -∞ ∈ ℝ* | |
6 | rexr 8002 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
7 | ltxr 9774 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))) | |
8 | 5, 6, 7 | sylancr 414 | . 2 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))) |
9 | 4, 8 | mpbird 167 | 1 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 class class class wbr 4003 ℝcr 7809 <ℝ cltrr 7814 +∞cpnf 7988 -∞cmnf 7989 ℝ*cxr 7990 < clt 7991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-cnex 7901 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-xp 4632 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 |
This theorem is referenced by: mnflt0 9783 mnfltxr 9785 xrlttr 9794 xrltso 9795 xrlttri3 9796 ngtmnft 9816 nmnfgt 9817 xrrebnd 9818 xrre3 9821 xltnegi 9834 xltadd1 9875 xposdif 9881 elico2 9936 elicc2 9937 ioomax 9947 elioomnf 9967 qbtwnxr 10257 tgioo 13982 |
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