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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 11309 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ≠ +∞) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 ℝcr 11154 +∞cpnf 11292 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-un 7755 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-nel 3047 df-rab 3437 df-v 3482 df-un 3956 df-in 3958 df-ss 3968 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 df-pnf 11297 |
| This theorem is referenced by: xaddnepnf 13279 dvfsumrlimge0 26071 dvfsumrlim 26072 dvfsumrlim2 26073 logno1 26678 fldextrspundgdvdslem 33730 limsupresico 45715 limsupvaluz2 45753 supcnvlimsup 45755 liminfresico 45786 xlimliminflimsup 45877 smflimsuplem2 46836 smflimsuplem5 46839 |
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