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Mirrors > Home > ILE Home > Th. List > renfdisj | GIF version |
Description: The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
renfdisj | ⊢ (ℝ ∩ {+∞, -∞}) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj 3411 | . 2 ⊢ ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞}) | |
2 | vex 2689 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | elpr 3548 | . . . 4 ⊢ (𝑥 ∈ {+∞, -∞} ↔ (𝑥 = +∞ ∨ 𝑥 = -∞)) |
4 | renepnf 7813 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ +∞) | |
5 | 4 | necon2bi 2363 | . . . . 5 ⊢ (𝑥 = +∞ → ¬ 𝑥 ∈ ℝ) |
6 | renemnf 7814 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞) | |
7 | 6 | necon2bi 2363 | . . . . 5 ⊢ (𝑥 = -∞ → ¬ 𝑥 ∈ ℝ) |
8 | 5, 7 | jaoi 705 | . . . 4 ⊢ ((𝑥 = +∞ ∨ 𝑥 = -∞) → ¬ 𝑥 ∈ ℝ) |
9 | 3, 8 | sylbi 120 | . . 3 ⊢ (𝑥 ∈ {+∞, -∞} → ¬ 𝑥 ∈ ℝ) |
10 | 9 | con2i 616 | . 2 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞}) |
11 | 1, 10 | mprgbir 2490 | 1 ⊢ (ℝ ∩ {+∞, -∞}) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ∩ cin 3070 ∅c0 3363 {cpr 3528 ℝcr 7619 +∞cpnf 7797 -∞cmnf 7798 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-pnf 7802 df-mnf 7803 |
This theorem is referenced by: (None) |
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