ltpnf - Intuitionistic Logic Explorer

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Theorem ltpnf 9988

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 2229	. . . 4 ⊢ +∞ = +∞
2		orc 717	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ +∞ = +∞) → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))
3	1, 2	mpan2 425	. . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))
4	3	olcd 739	. 2 ⊢ (𝐴 ∈ ℝ → ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <_ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))
5		rexr 8203	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
6		pnfxr 8210	. . 3 ⊢ +∞ ∈ ℝ^*
7		ltxr 9983	. . 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 713 = wceq 1395 ∈ wcel 2200 class class class wbr 4083 ℝcr 8009 <_ℝ cltrr 8014 +∞cpnf 8189 -∞cmnf 8190 ℝ^*cxr 8191 < clt 8192

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 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-cnex 8101

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 2801 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-xp 4725 df-pnf 8194 df-xr 8196 df-ltxr 8197

This theorem is referenced by: ltpnfd 9989 0ltpnf 9990 xrlttr 10003 xrltso 10004 xrlttri3 10005 nltpnft 10022 npnflt 10023 xrrebnd 10027 xrre 10028 xltnegi 10043 xltadd1 10084 xposdif 10090 elioc2 10144 elicc2 10146 ioomax 10156 ioopos 10158 elioopnf 10175 elicopnf 10177 qbtwnxr 10489 dfrp2 10495 filtinf 11025 xrmaxltsup 11784 fprodge0 12163 fprodge1 12165 xblss2ps 15093

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