mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 10195

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 2238	. . . 4 ⊢ -∞ = -∞
2		olc 723	. . . 4 ⊢ ((-∞ = -∞ ∧ 𝐴 ∈ ℝ) → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))
3	1, 2	mpan 428	. . 3 ⊢ (𝐴 ∈ ℝ → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))
4	3	olcd 746	. 2 ⊢ (𝐴 ∈ ℝ → ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))
5		mnfxr 8382	. . 3 ⊢ -∞ ∈ ℝ^*
6		rexr 8371	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
7		ltxr 10187	. . 3 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <_ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))))
8	5, 6, 7	sylancr 418	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 = wceq 1402 ∈ wcel 2209 class class class wbr 4130 ℝcr 8178 <_ℝ cltrr 8183 +∞cpnf 8357 -∞cmnf 8358 ℝ^*cxr 8359 < clt 8360

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-cnex 8270

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-xp 4780 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365

This theorem is used by: mnflt0 10196 mnfltxr 10198 xrlttr 10207 xrltso 10208 xrlttri3 10209 ngtmnft 10229 nmnfgt 10230 xrrebnd 10231 xrre3 10234 xltnegi 10247 xltadd1 10288 xposdif 10294 elico2 10349 elicc2 10350 ioomax 10360 elioomnf 10380 qbtwnxr 10702 tgioo 15704 repiecelem 17172 repiecele0 17173

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