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Mirrors > Home > MPE Home > Th. List > renemnf | Structured version Visualization version 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 10672 | . . . 4 ⊢ -∞ ∉ ℝ | |
2 | 1 | neli 3122 | . . 3 ⊢ ¬ -∞ ∈ ℝ |
3 | eleq1 2897 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ)) | |
4 | 2, 3 | mtbiri 328 | . 2 ⊢ (𝐴 = -∞ → ¬ 𝐴 ∈ ℝ) |
5 | 4 | necon2ai 3042 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ℝcr 10524 -∞cmnf 10661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 |
This theorem is referenced by: renemnfd 10681 renfdisj 10689 xrnemnf 12500 rexneg 12592 rexadd 12613 xaddnemnf 12617 xaddcom 12621 xaddid1 12622 xnegdi 12629 xpncan 12632 xleadd1a 12634 rexmul 12652 xadddilem 12675 xrs1mnd 20511 xrs10 20512 isxmet2d 22864 imasdsf1olem 22910 xaddeq0 30403 icorempo 34514 infrpge 41495 infleinflem1 41514 xrre4 41561 climxrre 41907 |
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