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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 2081 | . . . 4 ⊢ +∞ = +∞ | |
2 | orc 665 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ +∞ = +∞) → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))) | |
3 | 1, 2 | mpan2 415 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))) |
4 | 3 | olcd 685 | . 2 ⊢ (𝐴 ∈ ℝ → ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ)))) |
5 | rexr 7164 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
6 | pnfxr 8846 | . . 3 ⊢ +∞ ∈ ℝ* | |
7 | ltxr 8849 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))) | |
8 | 5, 6, 7 | sylancl 404 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < +∞ ↔ ((((𝐴 ∈ ℝ ∧ +∞ ∈ ℝ) ∧ 𝐴 <ℝ +∞) ∨ (𝐴 = -∞ ∧ +∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ +∞ = +∞) ∨ (𝐴 = -∞ ∧ +∞ ∈ ℝ))))) |
9 | 4, 8 | mpbird 165 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 661 = wceq 1284 ∈ wcel 1433 class class class wbr 3785 ℝcr 6980 <ℝ cltrr 6985 +∞cpnf 7150 -∞cmnf 7151 ℝ*cxr 7152 < clt 7153 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-cnex 7067 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-xr 7157 df-ltxr 7158 |
This theorem is referenced by: 0ltpnf 8857 xrlttr 8870 xrltso 8871 xrlttri3 8872 nltpnft 8884 xrrebnd 8886 xrre 8887 xltnegi 8902 elioc2 8959 elicc2 8961 ioomax 8971 ioopos 8973 elioopnf 8990 elicopnf 8992 qbtwnxr 9266 |
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