nltmnf - Metamath Proof Explorer

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Theorem nltmnf 11839

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 9961	. . . . . . 7 ⊢ -∞ ∉ ℝ
2	1	neli 2885	. . . . . 6 ⊢ ¬ -∞ ∈ ℝ
3	2	intnan 951	. . . . 5 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ ∈ ℝ)
4	3	intnanr 952	. . . 4 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞)
5		pnfnemnf 9973	. . . . . 6 ⊢ +∞ ≠ -∞
6	5	nesymi 2839	. . . . 5 ⊢ ¬ -∞ = +∞
7	6	intnan 951	. . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ = +∞)
8	4, 7	pm3.2ni 895	. . 3 ⊢ ¬ (((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞))
9	6	intnan 951	. . . 4 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ = +∞)
10	2	intnan 951	. . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ ∈ ℝ)
11	9, 10	pm3.2ni 895	. . 3 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))
12	8, 11	pm3.2ni 895	. 2 ⊢ ¬ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)))
13		mnfxr 9975	. . 3 ⊢ -∞ ∈ ℝ^*
14		ltxr 11825	. . 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 316	1 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → 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 ax-resscn 9872

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-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958

This theorem is referenced by: mnfle 11845 xrltnsym 11846 xrlttr 11849 qbtwnxr 11905 xltnegi 11921 xmullem2 11967 xmulasslem2 11984 xlemul1a 11990 xrsupexmnf 12007 xrsupsslem 12009 xrinfmsslem 12010 xrsup0 12025 reltxrnmnf 12043 infmremnf 12044 mnfnei 20835 blssioo 22406 deg1add 23667 icorempt2 32375 relowlssretop 32387 supxrgere 38490 supxrgelem 38494 infxrunb2 38525 iccpartiltu 39960

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