nltmnf - Metamath Proof Explorer

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Theorem nltmnf 13298

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 11391	. . . . . . 7 ⊢ -∞ ∉ ℝ
2	1	neli 3067	. . . . . 6 ⊢ ¬ -∞ ∈ ℝ
3	2	intnan 492	. . . . 5 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ ∈ ℝ)
4	3	intnanr 493	. . . 4 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞)
5		pnfnemnf 11403	. . . . . 6 ⊢ +∞ ≠ -∞
6	5	nesymi 3016	. . . . 5 ⊢ ¬ -∞ = +∞
7	6	intnan 492	. . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ = +∞)
8	4, 7	pm3.2ni 894	. . 3 ⊢ ¬ (((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞))
9	6	intnan 492	. . . 4 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ = +∞)
10	2	intnan 492	. . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ ∈ ℝ)
11	9, 10	pm3.2ni 894	. . 3 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))
12	8, 11	pm3.2ni 894	. 2 ⊢ ¬ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)))
13		mnfxr 11405	. . 3 ⊢ -∞ ∈ ℝ^*
14		ltxr 13284	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → (𝐴 < -∞ ↔ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)))))
15	13, 14	mpan2 704	. 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 class class class wbr 5108 ℝcr 11238 <_ℝ cltrr 11243 +∞cpnf 11379 -∞cmnf 11380 ℝ^*cxr 11381 < clt 11382

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387

This theorem is used by: mnfle 13304 xrltnsym 13306 xrlttr 13309 qbtwnxr 13370 xltnegi 13386 xmullem2 13435 xmulasslem2 13452 xlemul1a 13458 xrsupexmnf 13475 xrsupsslem 13477 xrinfmsslem 13478 xrsup0 13493 reltxrnmnf 13513 infmremnf 13514 mnfnei 23594 blssioo 25169 deg1add 26476 icorempo 38542 relowlssretop 38554 supxrgere 46618 supxrgelem 46622 infxrunb2 46652 iccpartiltu 48777

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