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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 7871 | . . . . . 6 ⊢ ℂ ∈ V | |
2 | 1 | uniex 4412 | . . . . 5 ⊢ ∪ ℂ ∈ V |
3 | pwuninel2 6244 | . . . . 5 ⊢ (∪ ℂ ∈ V → ¬ 𝒫 ∪ ℂ ∈ ℂ) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ |
5 | df-pnf 7929 | . . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ | |
6 | 5 | eleq1i 2230 | . . . 4 ⊢ (+∞ ∈ ℂ ↔ 𝒫 ∪ ℂ ∈ ℂ) |
7 | 4, 6 | mtbir 661 | . . 3 ⊢ ¬ +∞ ∈ ℂ |
8 | recn 7880 | . . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ) | |
9 | 7, 8 | mto 652 | . 2 ⊢ ¬ +∞ ∈ ℝ |
10 | 9 | nelir 2432 | 1 ⊢ +∞ ∉ ℝ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2135 ∉ wnel 2429 Vcvv 2724 𝒫 cpw 3556 ∪ cuni 3786 ℂcc 7745 ℝcr 7746 +∞cpnf 7924 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-un 4408 ax-cnex 7838 ax-resscn 7839 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-nel 2430 df-rex 2448 df-rab 2451 df-v 2726 df-in 3120 df-ss 3127 df-pw 3558 df-uni 3787 df-pnf 7929 |
This theorem is referenced by: renepnf 7940 nn0nepnf 9179 xrltnr 9709 pnfnlt 9717 xnn0lenn0nn0 9795 inftonninf 10370 pcgcd1 12253 pc2dvds 12255 |
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