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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 7768 | . . . . . 6 ⊢ ℂ ∈ V | |
2 | 1 | uniex 4367 | . . . . 5 ⊢ ∪ ℂ ∈ V |
3 | pwuninel2 6187 | . . . . 5 ⊢ (∪ ℂ ∈ V → ¬ 𝒫 ∪ ℂ ∈ ℂ) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ |
5 | df-pnf 7826 | . . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ | |
6 | 5 | eleq1i 2206 | . . . 4 ⊢ (+∞ ∈ ℂ ↔ 𝒫 ∪ ℂ ∈ ℂ) |
7 | 4, 6 | mtbir 661 | . . 3 ⊢ ¬ +∞ ∈ ℂ |
8 | recn 7777 | . . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ) | |
9 | 7, 8 | mto 652 | . 2 ⊢ ¬ +∞ ∈ ℝ |
10 | 9 | nelir 2407 | 1 ⊢ +∞ ∉ ℝ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1481 ∉ wnel 2404 Vcvv 2689 𝒫 cpw 3515 ∪ cuni 3744 ℂcc 7642 ℝcr 7643 +∞cpnf 7821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-un 4363 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-nel 2405 df-rex 2423 df-rab 2426 df-v 2691 df-in 3082 df-ss 3089 df-pw 3517 df-uni 3745 df-pnf 7826 |
This theorem is referenced by: renepnf 7837 nn0nepnf 9072 xrltnr 9596 pnfnlt 9603 xnn0lenn0nn0 9678 inftonninf 10245 |
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