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Mirrors > Home > MPE Home > Th. List > renepnfd | Structured version Visualization version GIF version |
Description: No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rexrd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
renepnfd | ⊢ (𝜑 → 𝐴 ≠ +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexrd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | renepnf 10678 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ≠ +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ≠ wne 2987 ℝcr 10525 +∞cpnf 10661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 df-uni 4801 df-pnf 10666 |
This theorem is referenced by: xaddnepnf 12618 dvfsumrlimge0 24633 dvfsumrlim 24634 dvfsumrlim2 24635 logno1 25227 limsupresico 42342 limsupvaluz2 42380 supcnvlimsup 42382 liminfresico 42413 xlimliminflimsup 42504 smflimsuplem2 43452 smflimsuplem5 43455 |
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