![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xnegex | Structured version Visualization version GIF version |
Description: A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegex | ⊢ -𝑒𝐴 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xneg 13092 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
2 | mnfxr 11271 | . . . 4 ⊢ -∞ ∈ ℝ* | |
3 | 2 | elexi 3494 | . . 3 ⊢ -∞ ∈ V |
4 | pnfex 11267 | . . . 4 ⊢ +∞ ∈ V | |
5 | negex 11458 | . . . 4 ⊢ -𝐴 ∈ V | |
6 | 4, 5 | ifex 4579 | . . 3 ⊢ if(𝐴 = -∞, +∞, -𝐴) ∈ V |
7 | 3, 6 | ifex 4579 | . 2 ⊢ if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) ∈ V |
8 | 1, 7 | eqeltri 2830 | 1 ⊢ -𝑒𝐴 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 ifcif 4529 +∞cpnf 11245 -∞cmnf 11246 ℝ*cxr 11247 -cneg 11445 -𝑒cxne 13089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-un 7725 ax-cnex 11166 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-uni 4910 df-iota 6496 df-fv 6552 df-ov 7412 df-pnf 11250 df-mnf 11251 df-xr 11252 df-neg 11447 df-xneg 13092 |
This theorem is referenced by: xrhmeo 24462 supminfxrrnmpt 44181 monoord2xrv 44194 liminfvalxr 44499 liminfpnfuz 44532 xlimpnfxnegmnf2 44574 |
Copyright terms: Public domain | W3C validator |