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Mirrors > Home > ILE Home > Th. List > pnfnre | 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 | cnex 7966 | . . . . . 6 ⊢ ℂ ∈ V | |
2 | 1 | uniex 4455 | . . . . 5 ⊢ ∪ ℂ ∈ V |
3 | pwuninel2 6308 | . . . . 5 ⊢ (∪ ℂ ∈ V → ¬ 𝒫 ∪ ℂ ∈ ℂ) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ |
5 | df-pnf 8025 | . . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ | |
6 | 5 | eleq1i 2255 | . . . 4 ⊢ (+∞ ∈ ℂ ↔ 𝒫 ∪ ℂ ∈ ℂ) |
7 | 4, 6 | mtbir 672 | . . 3 ⊢ ¬ +∞ ∈ ℂ |
8 | recn 7975 | . . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ) | |
9 | 7, 8 | mto 663 | . 2 ⊢ ¬ +∞ ∈ ℝ |
10 | 9 | nelir 2458 | 1 ⊢ +∞ ∉ ℝ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2160 ∉ wnel 2455 Vcvv 2752 𝒫 cpw 3590 ∪ cuni 3824 ℂcc 7840 ℝcr 7841 +∞cpnf 8020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-un 4451 ax-cnex 7933 ax-resscn 7934 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-nel 2456 df-rex 2474 df-rab 2477 df-v 2754 df-in 3150 df-ss 3157 df-pw 3592 df-uni 3825 df-pnf 8025 |
This theorem is referenced by: renepnf 8036 nn0nepnf 9278 xrltnr 9811 pnfnlt 9819 xnn0lenn0nn0 9897 inftonninf 10474 pcgcd1 12363 pc2dvds 12365 |
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