ltpnf - Intuitionistic Logic Explorer

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Theorem ltpnf 10161

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

**Assertion**
Ref	Expression
ltpnf	⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)

**Proof of Theorem ltpnf**
Step	Hyp	Ref	Expression
1		eqid 2238	. . . 4 ⊢ +∞ = +∞
2		orc 724	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ +∞ = +∞) → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))
3	1, 2	mpan2 429	. . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))
4	3	olcd 746	. 2 ⊢ (𝐴 ∈ ℝ → ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <_ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))
5		rexr 8361	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
6		pnfxr 8368	. . 3 ⊢ +∞ ∈ ℝ^*
7		ltxr 10156	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <_ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))))
8	5, 6, 7	sylancl 417	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <_ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (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 720 = wceq 1402 ∈ wcel 2209 class class class wbr 4125 ℝcr 8168 <_ℝ cltrr 8173 +∞cpnf 8347 -∞cmnf 8348 ℝ^*cxr 8349 < clt 8350

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 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 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-cnex 8260

This theorem 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 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-xp 4775 df-pnf 8352 df-xr 8354 df-ltxr 8355

This theorem is referenced by: ltpnfd 10162 0ltpnf 10163 xrlttr 10176 xrltso 10177 xrlttri3 10178 nltpnft 10195 npnflt 10196 xrrebnd 10200 xrre 10201 xltnegi 10216 xltadd1 10257 xposdif 10263 elioc2 10317 elicc2 10319 ioomax 10329 ioopos 10331 elioopnf 10348 elicopnf 10350 qbtwnxr 10670 dfrp2 10676 filtinf 11208 xrmaxltsup 12002 fprodge0 12382 fprodge1 12384 xblss2ps 15428

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