mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 10008

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 2229	. . . 4 ⊢ -∞ = -∞
2		olc 716	. . . 4 ⊢ ((-∞ = -∞ ∧ 𝐴 ∈ ℝ) → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))
3	1, 2	mpan 424	. . 3 ⊢ (𝐴 ∈ ℝ → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))
4	3	olcd 739	. 2 ⊢ (𝐴 ∈ ℝ → ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))
5		mnfxr 8226	. . 3 ⊢ -∞ ∈ ℝ^*
6		rexr 8215	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
7		ltxr 10000	. . 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 713 = wceq 1395 ∈ wcel 2200 class class class wbr 4086 ℝcr 8021 <_ℝ cltrr 8026 +∞cpnf 8201 -∞cmnf 8202 ℝ^*cxr 8203 < clt 8204

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 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-cnex 8113

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-v 2802 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-xp 4729 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209

This theorem is referenced by: mnflt0 10009 mnfltxr 10011 xrlttr 10020 xrltso 10021 xrlttri3 10022 ngtmnft 10042 nmnfgt 10043 xrrebnd 10044 xrre3 10047 xltnegi 10060 xltadd1 10101 xposdif 10107 elico2 10162 elicc2 10163 ioomax 10173 elioomnf 10193 qbtwnxr 10507 tgioo 15268

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