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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 3463 | . 2 ⊢ ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞}) | |
2 | vex 2733 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | elpr 3604 | . . . 4 ⊢ (𝑥 ∈ {+∞, -∞} ↔ (𝑥 = +∞ ∨ 𝑥 = -∞)) |
4 | renepnf 7967 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ +∞) | |
5 | 4 | necon2bi 2395 | . . . . 5 ⊢ (𝑥 = +∞ → ¬ 𝑥 ∈ ℝ) |
6 | renemnf 7968 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞) | |
7 | 6 | necon2bi 2395 | . . . . 5 ⊢ (𝑥 = -∞ → ¬ 𝑥 ∈ ℝ) |
8 | 5, 7 | jaoi 711 | . . . 4 ⊢ ((𝑥 = +∞ ∨ 𝑥 = -∞) → ¬ 𝑥 ∈ ℝ) |
9 | 3, 8 | sylbi 120 | . . 3 ⊢ (𝑥 ∈ {+∞, -∞} → ¬ 𝑥 ∈ ℝ) |
10 | 9 | con2i 622 | . 2 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞}) |
11 | 1, 10 | mprgbir 2528 | 1 ⊢ (ℝ ∩ {+∞, -∞}) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∨ wo 703 = wceq 1348 ∈ wcel 2141 ∩ cin 3120 ∅c0 3414 {cpr 3584 ℝcr 7773 +∞cpnf 7951 -∞cmnf 7952 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-pnf 7956 df-mnf 7957 |
This theorem is referenced by: (None) |
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