ltpnf - Intuitionistic Logic Explorer

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Theorem ltpnf 9855

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 2196	. . . 4 ⊢ +∞ = +∞
2		orc 713	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ +∞ = +∞) → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))
3	1, 2	mpan2 425	. . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))
4	3	olcd 735	. 2 ⊢ (𝐴 ∈ ℝ → ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <_ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))
5		rexr 8072	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
6		pnfxr 8079	. . 3 ⊢ +∞ ∈ ℝ^*
7		ltxr 9850	. . 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 709 = wceq 1364 ∈ wcel 2167 class class class wbr 4033 ℝcr 7878 <_ℝ cltrr 7883 +∞cpnf 8058 -∞cmnf 8059 ℝ^*cxr 8060 < clt 8061

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-cnex 7970

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-v 2765 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-xp 4669 df-pnf 8063 df-xr 8065 df-ltxr 8066

This theorem is referenced by: ltpnfd 9856 0ltpnf 9857 xrlttr 9870 xrltso 9871 xrlttri3 9872 nltpnft 9889 npnflt 9890 xrrebnd 9894 xrre 9895 xltnegi 9910 xltadd1 9951 xposdif 9957 elioc2 10011 elicc2 10013 ioomax 10023 ioopos 10025 elioopnf 10042 elicopnf 10044 qbtwnxr 10347 dfrp2 10353 filtinf 10883 xrmaxltsup 11423 fprodge0 11802 fprodge1 11804 xblss2ps 14640

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