Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ltpnf | GIF version |
Description: Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
ltpnf | ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2170 | . . . 4 ⊢ +∞ = +∞ | |
2 | orc 707 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ +∞ = +∞) → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))) | |
3 | 1, 2 | mpan2 423 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))) |
4 | 3 | olcd 729 | . 2 ⊢ (𝐴 ∈ ℝ → ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))) |
5 | rexr 7965 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
6 | pnfxr 7972 | . . 3 ⊢ +∞ ∈ ℝ* | |
7 | ltxr 9732 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))) | |
8 | 5, 6, 7 | sylancl 411 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))) |
9 | 4, 8 | mpbird 166 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 703 = wceq 1348 ∈ wcel 2141 class class class wbr 3989 ℝcr 7773 <ℝ cltrr 7778 +∞cpnf 7951 -∞cmnf 7952 ℝ*cxr 7953 < clt 7954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-cnex 7865 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-xr 7958 df-ltxr 7959 |
This theorem is referenced by: ltpnfd 9738 0ltpnf 9739 xrlttr 9752 xrltso 9753 xrlttri3 9754 nltpnft 9771 npnflt 9772 xrrebnd 9776 xrre 9777 xltnegi 9792 xltadd1 9833 xposdif 9839 elioc2 9893 elicc2 9895 ioomax 9905 ioopos 9907 elioopnf 9924 elicopnf 9926 qbtwnxr 10214 dfrp2 10220 filtinf 10726 xrmaxltsup 11221 fprodge0 11600 fprodge1 11602 xblss2ps 13198 |
Copyright terms: Public domain | W3C validator |