nltmnf - Metamath Proof Explorer

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Theorem nltmnf 13155

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 11253	. . . . . . 7 ⊢ -∞ ∉ ℝ
2	1	neli 3066	. . . . . 6 ⊢ ¬ -∞ ∈ ℝ
3	2	intnan 491	. . . . 5 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ ∈ ℝ)
4	3	intnanr 492	. . . 4 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞)
5		pnfnemnf 11265	. . . . . 6 ⊢ +∞ ≠ -∞
6	5	nesymi 3015	. . . . 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 11267	. . 3 ⊢ -∞ ∈ ℝ^*
14		ltxr 13141	. . 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 1570 ∈ wcel 2143 class class class wbr 5110 ℝcr 11100 <_ℝ cltrr 11105 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 < clt 11244

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249

This theorem is referenced by: mnfle 13161 xrltnsym 13163 xrlttr 13166 qbtwnxr 13227 xltnegi 13243 xmullem2 13292 xmulasslem2 13309 xlemul1a 13315 xrsupexmnf 13332 xrsupsslem 13334 xrinfmsslem 13335 xrsup0 13350 reltxrnmnf 13370 infmremnf 13371 mnfnei 23359 blssioo 24933 deg1add 26241 icorempo 37978 relowlssretop 37990 supxrgere 46032 supxrgelem 46036 infxrunb2 46066 iccpartiltu 48154

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