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Theorem mnflt 11833

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 2610	. . . 4 ⊢ -∞ = -∞
2		olc 398	. . . 4 ⊢ ((-∞ = -∞ ∧ 𝐴 ∈ ℝ) → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))
3	1, 2	mpan 702	. . 3 ⊢ (𝐴 ∈ ℝ → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))
4	3	olcd 407	. 2 ⊢ (𝐴 ∈ ℝ → ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))
5		mnfxr 9975	. . 3 ⊢ -∞ ∈ ℝ^*
6		rexr 9964	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
7		ltxr 11825	. . 3 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))))
8	5, 6, 7	sylancr 694	. 2 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))))
9	4, 8	mpbird 246	1 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ℝcr 9814 <_ℝ cltrr 9819 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958

This theorem is referenced by: mnfltd 11834 mnflt0 11835 mnfltxr 11837 xrlttri 11848 xrlttr 11849 xrrebnd 11873 xrre3 11876 qbtwnxr 11905 xltnegi 11921 xrsupsslem 12009 xrub 12014 supxrre 12029 elico2 12108 elicc2 12109 ioomax 12119 elioomnf 12139 difreicc 12175 icopnfcld 22381 iocmnfcld 22382 tgioo 22407 xrtgioo 22417 reconnlem1 22437 reconnlem2 22438 bndth 22565 ovoliunlem1 23077 ovoliun 23080 ioombl1lem2 23134 mbfmax 23222 ismbf3d 23227 itg2seq 23315 dvferm1lem 23551 dvferm2lem 23553 degltlem1 23636 ply1divex 23700 dvdsq1p 23724 ellogdm 24185 logdmnrp 24187 atans2 24458 esumcvgsum 29477 dya2iocbrsiga 29664 dya2icobrsiga 29665 orvclteel 29861 iooelexlt 32386 itg2addnclem 32631 asindmre 32665 dvasin 32666 dvacos 32667 rfcnpre4 38216 infrpge 38508 infxr 38524 infxrunb2 38525 infleinflem2 38528 icccncfext 38773 fourierdlem113 39112 fouriersw 39124 iccpartigtl 39961

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