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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 3486 | . 2 ⊢ ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞}) | |
2 | vex 2755 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | elpr 3628 | . . . 4 ⊢ (𝑥 ∈ {+∞, -∞} ↔ (𝑥 = +∞ ∨ 𝑥 = -∞)) |
4 | renepnf 8035 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ +∞) | |
5 | 4 | necon2bi 2415 | . . . . 5 ⊢ (𝑥 = +∞ → ¬ 𝑥 ∈ ℝ) |
6 | renemnf 8036 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞) | |
7 | 6 | necon2bi 2415 | . . . . 5 ⊢ (𝑥 = -∞ → ¬ 𝑥 ∈ ℝ) |
8 | 5, 7 | jaoi 717 | . . . 4 ⊢ ((𝑥 = +∞ ∨ 𝑥 = -∞) → ¬ 𝑥 ∈ ℝ) |
9 | 3, 8 | sylbi 121 | . . 3 ⊢ (𝑥 ∈ {+∞, -∞} → ¬ 𝑥 ∈ ℝ) |
10 | 9 | con2i 628 | . 2 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞}) |
11 | 1, 10 | mprgbir 2548 | 1 ⊢ (ℝ ∩ {+∞, -∞}) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∨ wo 709 = wceq 1364 ∈ wcel 2160 ∩ cin 3143 ∅c0 3437 {cpr 3608 ℝcr 7840 +∞cpnf 8019 -∞cmnf 8020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-pnf 8024 df-mnf 8025 |
This theorem is referenced by: (None) |
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