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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 10683 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ≠ +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ≠ wne 3016 ℝcr 10530 +∞cpnf 10666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-pw 4540 df-sn 4561 df-pr 4563 df-uni 4832 df-pnf 10671 |
This theorem is referenced by: xaddnepnf 12624 dvfsumrlimge0 24621 dvfsumrlim 24622 dvfsumrlim2 24623 logno1 25213 limsupresico 41974 limsupvaluz2 42012 supcnvlimsup 42014 liminfresico 42045 xlimliminflimsup 42136 smflimsuplem2 43089 smflimsuplem5 43092 |
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