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Theorem mnflt 11957

Description: Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
mnflt	⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)

**Proof of Theorem mnflt**
Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 ⊢ -∞ = -∞
2		olc 399	. . . 4 ⊢ ((-∞ = -∞ ∧ 𝐴 ∈ ℝ) → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))
3	1, 2	mpan 706	. . 3 ⊢ (𝐴 ∈ ℝ → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))
4	3	olcd 408	. 2 ⊢ (𝐴 ∈ ℝ → ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))
5		mnfxr 10096	. . 3 ⊢ -∞ ∈ ℝ^*
6		rexr 10085	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
7		ltxr 11949	. . 3 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))))
8	5, 6, 7	sylancr 695	. 2 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))))
9	4, 8	mpbird 247	1 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ℝcr 9935 <_ℝ cltrr 9940 +∞cpnf 10071 -∞cmnf 10072 ℝ^*cxr 10073 < clt 10074

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079

This theorem is referenced by: mnfltd 11958 mnflt0 11959 mnfltxr 11961 xrlttri 11972 xrlttr 11973 xrrebnd 11999 xrre3 12002 qbtwnxr 12031 xltnegi 12047 xrsupsslem 12137 xrub 12142 supxrre 12157 elico2 12237 elicc2 12238 ioomax 12248 elioomnf 12268 difreicc 12304 icopnfcld 22571 iocmnfcld 22572 tgioo 22599 xrtgioo 22609 reconnlem1 22629 reconnlem2 22630 bndth 22757 ovoliunlem1 23270 ovoliun 23273 ioombl1lem2 23327 mbfmax 23416 ismbf3d 23421 itg2seq 23509 dvferm1lem 23747 dvferm2lem 23749 degltlem1 23832 ply1divex 23896 dvdsq1p 23920 ellogdm 24385 logdmnrp 24387 atans2 24658 esumcvgsum 30150 dya2iocbrsiga 30337 dya2icobrsiga 30338 orvclteel 30534 iooelexlt 33210 itg2addnclem 33461 asindmre 33495 dvasin 33496 dvacos 33497 rfcnpre4 39193 infrpge 39567 infxr 39583 infxrunb2 39584 infleinflem2 39587 icccncfext 40100 fourierdlem113 40436 fouriersw 40448 iccpartigtl 41359

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