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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 7068 | . . . 4 ⊢ -∞ ∉ ℝ | |
2 | 1 | neli 2299 | . . 3 ⊢ ¬ -∞ ∈ ℝ |
3 | eleq1 2100 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ)) | |
4 | 2, 3 | mtbiri 600 | . 2 ⊢ (𝐴 = -∞ → ¬ 𝐴 ∈ ℝ) |
5 | 4 | necon2ai 2259 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 ≠ wne 2204 ℝcr 6888 -∞cmnf 7058 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-pnf 7062 df-mnf 7063 |
This theorem is referenced by: renemnfd 7077 renfdisj 7079 ltxrlt 7085 xrnemnf 8699 xrlttri3 8718 ngtmnft 8731 xrrebnd 8732 rexneg 8743 |
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