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Mirrors > Home > ILE Home > Th. List > renemnf | GIF version |
Description: No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
renemnf | ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfnre 7451 | . . . 4 ⊢ -∞ ∉ ℝ | |
2 | 1 | neli 2348 | . . 3 ⊢ ¬ -∞ ∈ ℝ |
3 | eleq1 2147 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ)) | |
4 | 2, 3 | mtbiri 633 | . 2 ⊢ (𝐴 = -∞ → ¬ 𝐴 ∈ ℝ) |
5 | 4 | necon2ai 2305 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1287 ∈ wcel 1436 ≠ wne 2251 ℝcr 7270 -∞cmnf 7441 |
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 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-setind 4319 ax-cnex 7357 ax-resscn 7358 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-v 2616 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-uni 3631 df-pnf 7445 df-mnf 7446 |
This theorem is referenced by: renemnfd 7460 renfdisj 7467 ltxrlt 7473 xrnemnf 9157 xrlttri3 9176 ngtmnft 9189 xrrebnd 9190 rexneg 9201 |
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