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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 10767 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ≠ +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 2934 ℝcr 10614 +∞cpnf 10750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 ax-resscn 10672 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-nel 3039 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-pw 4490 df-sn 4517 df-pr 4519 df-uni 4797 df-pnf 10755 |
This theorem is referenced by: xaddnepnf 12713 dvfsumrlimge0 24782 dvfsumrlim 24783 dvfsumrlim2 24784 logno1 25379 limsupresico 42783 limsupvaluz2 42821 supcnvlimsup 42823 liminfresico 42854 xlimliminflimsup 42945 smflimsuplem2 43893 smflimsuplem5 43896 |
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