mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 10116

Description: Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
mnflt	⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)

**Proof of Theorem mnflt**
Step	Hyp	Ref	Expression
1		eqid 2232	. . . 4 ⊢ -∞ = -∞
2		olc 719	. . . 4 ⊢ ((-∞ = -∞ ∧ 𝐴 ∈ ℝ) → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))
3	1, 2	mpan 424	. . 3 ⊢ (𝐴 ∈ ℝ → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))
4	3	olcd 742	. 2 ⊢ (𝐴 ∈ ℝ → ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))
5		mnfxr 8330	. . 3 ⊢ -∞ ∈ ℝ^*
6		rexr 8319	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
7		ltxr 10108	. . 3 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))))
8	5, 6, 7	sylancr 414	. 2 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))))
9	4, 8	mpbird 167	1 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 = wceq 1398 ∈ wcel 2203 class class class wbr 4109 ℝcr 8126 <_ℝ cltrr 8131 +∞cpnf 8305 -∞cmnf 8306 ℝ^*cxr 8307 < clt 8308

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-cnex 8218

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-v 2815 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-xp 4755 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313

This theorem is referenced by: mnflt0 10117 mnfltxr 10119 xrlttr 10128 xrltso 10129 xrlttri3 10130 ngtmnft 10150 nmnfgt 10151 xrrebnd 10152 xrre3 10155 xltnegi 10168 xltadd1 10209 xposdif 10215 elico2 10270 elicc2 10271 ioomax 10281 elioomnf 10301 qbtwnxr 10617 tgioo 15419 repiecelem 16809 repiecele0 16810

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