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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 7932 | . . . 4 ⊢ -∞ ∉ ℝ | |
2 | 1 | neli 2431 | . . 3 ⊢ ¬ -∞ ∈ ℝ |
3 | eleq1 2227 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ)) | |
4 | 2, 3 | mtbiri 665 | . 2 ⊢ (𝐴 = -∞ → ¬ 𝐴 ∈ ℝ) |
5 | 4 | necon2ai 2388 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 ℝcr 7743 -∞cmnf 7922 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-pnf 7926 df-mnf 7927 |
This theorem is referenced by: renemnfd 7941 renfdisj 7949 ltxrlt 7955 xrnemnf 9704 xrlttri3 9724 ngtmnft 9744 xrrebnd 9746 rexneg 9757 xrmnfdc 9770 rexadd 9779 xaddnemnf 9784 xaddcom 9788 xaddid1 9789 xnegdi 9795 xpncan 9798 xleadd1a 9800 xltadd1 9803 xposdif 9809 xrmaxrecl 11182 isxmet2d 12895 |
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