ltpnf - Intuitionistic Logic Explorer

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Theorem ltpnf 10132

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 2234	. . . 4 ⊢ +∞ = +∞
2		orc 720	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ +∞ = +∞) → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))
3	1, 2	mpan2 425	. . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))
4	3	olcd 742	. 2 ⊢ (𝐴 ∈ ℝ → ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <_ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))
5		rexr 8335	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
6		pnfxr 8342	. . 3 ⊢ +∞ ∈ ℝ^*
7		ltxr 10127	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <_ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))))
8	5, 6, 7	sylancl 413	. 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 716 = wceq 1398 ∈ wcel 2205 class class class wbr 4114 ℝcr 8142 <_ℝ cltrr 8147 +∞cpnf 8321 -∞cmnf 8322 ℝ^*cxr 8323 < clt 8324

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 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-cnex 8234

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ral 2527 df-rex 2528 df-v 2817 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-xp 4760 df-pnf 8326 df-xr 8328 df-ltxr 8329

This theorem is referenced by: ltpnfd 10133 0ltpnf 10134 xrlttr 10147 xrltso 10148 xrlttri3 10149 nltpnft 10166 npnflt 10167 xrrebnd 10171 xrre 10172 xltnegi 10187 xltadd1 10228 xposdif 10234 elioc2 10288 elicc2 10290 ioomax 10300 ioopos 10302 elioopnf 10319 elicopnf 10321 qbtwnxr 10641 dfrp2 10647 filtinf 11179 xrmaxltsup 11968 fprodge0 12348 fprodge1 12350 xblss2ps 15395

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