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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 9452 | . 2 ⊢ -𝑒-∞ = if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) | |
2 | mnfnepnf 7745 | . . 3 ⊢ -∞ ≠ +∞ | |
3 | ifnefalse 3451 | . . 3 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞)) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ if(-∞ = +∞, -∞, if(-∞ = -∞, +∞, --∞)) = if(-∞ = -∞, +∞, --∞) |
5 | eqid 2115 | . . 3 ⊢ -∞ = -∞ | |
6 | 5 | iftruei 3446 | . 2 ⊢ if(-∞ = -∞, +∞, --∞) = +∞ |
7 | 1, 4, 6 | 3eqtri 2139 | 1 ⊢ -𝑒-∞ = +∞ |
Colors of variables: wff set class |
Syntax hints: = wceq 1314 ≠ wne 2282 ifcif 3440 +∞cpnf 7721 -∞cmnf 7722 -cneg 7857 -𝑒cxne 9449 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-un 4315 ax-cnex 7636 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-rex 2396 df-rab 2399 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-if 3441 df-pw 3478 df-sn 3499 df-pr 3500 df-uni 3703 df-pnf 7726 df-mnf 7727 df-xr 7728 df-xneg 9452 |
This theorem is referenced by: xnegcl 9508 xnegneg 9509 xltnegi 9511 xnegid 9535 xnegdi 9544 xsubge0 9557 xposdif 9558 |
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