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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 8031 | . . . 4 ⊢ -∞ ∉ ℝ | |
2 | 1 | neli 2457 | . . 3 ⊢ ¬ -∞ ∈ ℝ |
3 | eleq1 2252 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ)) | |
4 | 2, 3 | mtbiri 676 | . 2 ⊢ (𝐴 = -∞ → ¬ 𝐴 ∈ ℝ) |
5 | 4 | necon2ai 2414 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ℝcr 7841 -∞cmnf 8021 |
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-ext 2171 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 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-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-pnf 8025 df-mnf 8026 |
This theorem is referenced by: renemnfd 8040 renfdisj 8048 ltxrlt 8054 xrnemnf 9809 xrlttri3 9829 ngtmnft 9849 xrrebnd 9851 rexneg 9862 xrmnfdc 9875 rexadd 9884 xaddnemnf 9889 xaddcom 9893 xaddid1 9894 xnegdi 9900 xpncan 9903 xleadd1a 9905 xltadd1 9908 xposdif 9914 xrmaxrecl 11298 isxmet2d 14325 |
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