nltmnf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > nltmnf		Structured version Visualization version GIF version

Theorem nltmnf 13055

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 11187	. . . . . . 7 ⊢ -∞ ∉ ℝ
2	1	neli 3039	. . . . . 6 ⊢ ¬ -∞ ∈ ℝ
3	2	intnan 486	. . . . 5 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ ∈ ℝ)
4	3	intnanr 487	. . . 4 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞)
5		pnfnemnf 11199	. . . . . 6 ⊢ +∞ ≠ -∞
6	5	nesymi 2990	. . . . 5 ⊢ ¬ -∞ = +∞
7	6	intnan 486	. . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ = +∞)
8	4, 7	pm3.2ni 881	. . 3 ⊢ ¬ (((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞))
9	6	intnan 486	. . . 4 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ = +∞)
10	2	intnan 486	. . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ ∈ ℝ)
11	9, 10	pm3.2ni 881	. . 3 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))
12	8, 11	pm3.2ni 881	. 2 ⊢ ¬ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)))
13		mnfxr 11201	. . 3 ⊢ -∞ ∈ ℝ^*
14		ltxr 13041	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → (𝐴 < -∞ ↔ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)))))
15	13, 14	mpan2 692	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐴 < -∞ ↔ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <_ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)))))
16	12, 15	mtbiri 327	1 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 class class class wbr 5100 ℝcr 11037 <_ℝ cltrr 11042 +∞cpnf 11175 -∞cmnf 11176 ℝ^*cxr 11177 < clt 11178

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183

This theorem is referenced by: mnfle 13061 xrltnsym 13063 xrlttr 13066 qbtwnxr 13127 xltnegi 13143 xmullem2 13192 xmulasslem2 13209 xlemul1a 13215 xrsupexmnf 13232 xrsupsslem 13234 xrinfmsslem 13235 xrsup0 13250 reltxrnmnf 13270 infmremnf 13271 mnfnei 23177 blssioo 24751 deg1add 26076 icorempo 37603 relowlssretop 37615 supxrgere 45689 supxrgelem 45693 infxrunb2 45723 iccpartiltu 47779

W3C validator