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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 3292 | . 2 ⊢ ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞}) | |
2 | vex 2604 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | elpr 3419 | . . . 4 ⊢ (𝑥 ∈ {+∞, -∞} ↔ (𝑥 = +∞ ∨ 𝑥 = -∞)) |
4 | renepnf 7166 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ +∞) | |
5 | 4 | necon2bi 2300 | . . . . 5 ⊢ (𝑥 = +∞ → ¬ 𝑥 ∈ ℝ) |
6 | renemnf 7167 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞) | |
7 | 6 | necon2bi 2300 | . . . . 5 ⊢ (𝑥 = -∞ → ¬ 𝑥 ∈ ℝ) |
8 | 5, 7 | jaoi 668 | . . . 4 ⊢ ((𝑥 = +∞ ∨ 𝑥 = -∞) → ¬ 𝑥 ∈ ℝ) |
9 | 3, 8 | sylbi 119 | . . 3 ⊢ (𝑥 ∈ {+∞, -∞} → ¬ 𝑥 ∈ ℝ) |
10 | 9 | con2i 589 | . 2 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞}) |
11 | 1, 10 | mprgbir 2421 | 1 ⊢ (ℝ ∩ {+∞, -∞}) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∨ wo 661 = wceq 1284 ∈ wcel 1433 ∩ cin 2972 ∅c0 3251 {cpr 3399 ℝcr 6980 +∞cpnf 7150 -∞cmnf 7151 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-pnf 7155 df-mnf 7156 |
This theorem is referenced by: (None) |
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