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Mirrors > Home > ILE Home > Th. List > xnegmnf | GIF version |
Description: Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegmnf | ⊢ -𝑒-∞ = +∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xneg 9559 | . 2 ⊢ -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) | |
2 | mnfnepnf 7821 | . . 3 ⊢ -∞ ≠ +∞ | |
3 | ifnefalse 3485 | . . 3 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞) |
5 | eqid 2139 | . . 3 ⊢ -∞ = -∞ | |
6 | 5 | iftruei 3480 | . 2 ⊢ if(-∞ = -∞, +∞, --∞) = +∞ |
7 | 1, 4, 6 | 3eqtri 2164 | 1 ⊢ -𝑒-∞ = +∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ≠ wne 2308 ifcif 3474 +∞cpnf 7797 -∞cmnf 7798 -cneg 7934 -𝑒cxne 9556 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-un 4355 ax-cnex 7711 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-pnf 7802 df-mnf 7803 df-xr 7804 df-xneg 9559 |
This theorem is referenced by: xnegcl 9615 xnegneg 9616 xltnegi 9618 xnegid 9642 xnegdi 9651 xsubge0 9664 xposdif 9665 |
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