nltmnf - Metamath Proof Explorer

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Theorem nltmnf 13150

Description: No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)

**Assertion**
Ref	Expression
nltmnf	⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)

**Proof of Theorem nltmnf**
Step	Hyp	Ref	Expression
1		mnfnre 11248	. . . . . . 7 ⊢ -∞ ∉ ℝ
2	1	neli 3072	. . . . . 6 ⊢ ¬ -∞ ∈ ℝ
3	2	intnan 491	. . . . 5 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ ∈ ℝ)
4	3	intnanr 492	. . . 4 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞)
5		pnfnemnf 11260	. . . . . 6 ⊢ +∞ ≠ -∞
6	5	nesymi 3021	. . . . 5 ⊢ ¬ -∞ = +∞
7	6	intnan 491	. . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ = +∞)
8	4, 7	pm3.2ni 893	. . 3 ⊢ ¬ (((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞))
9	6	intnan 491	. . . 4 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ = +∞)
10	2	intnan 491	. . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ ∈ ℝ)
11	9, 10	pm3.2ni 893	. . 3 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))
12	8, 11	pm3.2ni 893	. 2 ⊢ ¬ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)))
13		mnfxr 11262	. . 3 ⊢ -∞ ∈ ℝ^*
14		ltxr 13136	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → (𝐴 < -∞ ↔ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)))))
15	13, 14	mpan2 703	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐴 < -∞ ↔ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)))))
16	12, 15	mtbiri 330	1 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1567 ∈ wcel 2149 class class class wbr 5110 ℝcr 11095 <_ℝ cltrr 11100 +∞cpnf 11236 -∞cmnf 11237 ℝ^*cxr 11238 < clt 11239

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244

This theorem is referenced by: mnfle 13156 xrltnsym 13158 xrlttr 13161 qbtwnxr 13222 xltnegi 13238 xmullem2 13287 xmulasslem2 13304 xlemul1a 13310 xrsupexmnf 13327 xrsupsslem 13329 xrinfmsslem 13330 xrsup0 13345 reltxrnmnf 13365 infmremnf 13366 mnfnei 23343 blssioo 24917 deg1add 26225 icorempo 37880 relowlssretop 37892 supxrgere 45934 supxrgelem 45938 infxrunb2 45968 iccpartiltu 48053

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