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| Mirrors > Home > MPE Home > Th. List > pnfnre | Structured version Visualization version GIF version | ||
| Description: Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
| Ref | Expression |
|---|---|
| pnfnre | ⊢ +∞ ∉ ℝ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-pnf 11297 | . . . 4 ⊢ +∞ = 𝒫 ∪ ℂ | |
| 2 | pwuninel 8300 | . . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ | |
| 3 | 1, 2 | eqneltri 2860 | . . 3 ⊢ ¬ +∞ ∈ ℂ |
| 4 | recn 11245 | . . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ) | |
| 5 | 3, 4 | mto 197 | . 2 ⊢ ¬ +∞ ∈ ℝ |
| 6 | 5 | nelir 3049 | 1 ⊢ +∞ ∉ ℝ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∉ wnel 3046 𝒫 cpw 4600 ∪ cuni 4907 ℂcc 11153 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-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: pnfnre2 11303 renepnf 11309 ltxrlt 11331 nn0nepnf 12607 xrltnr 13161 pnfnlt 13170 xnn0lenn0nn0 13287 hashclb 14397 hasheq0 14402 pcgcd1 16915 pc2dvds 16917 ramtcl2 17049 odhash3 19594 xrsdsreclblem 21430 pnfnei 23228 iccpnfcnv 24975 i1f0rn 25717 |
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