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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 3416 | . 2 ⊢ ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞}) | |
2 | vex 2692 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | elpr 3553 | . . . 4 ⊢ (𝑥 ∈ {+∞, -∞} ↔ (𝑥 = +∞ ∨ 𝑥 = -∞)) |
4 | renepnf 7837 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ +∞) | |
5 | 4 | necon2bi 2364 | . . . . 5 ⊢ (𝑥 = +∞ → ¬ 𝑥 ∈ ℝ) |
6 | renemnf 7838 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞) | |
7 | 6 | necon2bi 2364 | . . . . 5 ⊢ (𝑥 = -∞ → ¬ 𝑥 ∈ ℝ) |
8 | 5, 7 | jaoi 706 | . . . 4 ⊢ ((𝑥 = +∞ ∨ 𝑥 = -∞) → ¬ 𝑥 ∈ ℝ) |
9 | 3, 8 | sylbi 120 | . . 3 ⊢ (𝑥 ∈ {+∞, -∞} → ¬ 𝑥 ∈ ℝ) |
10 | 9 | con2i 617 | . 2 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞}) |
11 | 1, 10 | mprgbir 2493 | 1 ⊢ (ℝ ∩ {+∞, -∞}) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∨ wo 698 = wceq 1332 ∈ wcel 1481 ∩ cin 3075 ∅c0 3368 {cpr 3533 ℝcr 7643 +∞cpnf 7821 -∞cmnf 7822 |
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-setind 4460 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-3an 965 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-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-pnf 7826 df-mnf 7827 |
This theorem is referenced by: (None) |
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