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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 13041 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
2 | mnfxr 11220 | . . . 4 ⊢ -∞ ∈ ℝ* | |
3 | 2 | elexi 3466 | . . 3 ⊢ -∞ ∈ V |
4 | pnfex 11216 | . . . 4 ⊢ +∞ ∈ V | |
5 | negex 11407 | . . . 4 ⊢ -𝐴 ∈ V | |
6 | 4, 5 | ifex 4540 | . . 3 ⊢ if(𝐴 = -∞, +∞, -𝐴) ∈ V |
7 | 3, 6 | ifex 4540 | . 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 3447 ifcif 4490 +∞cpnf 11194 -∞cmnf 11195 ℝ*cxr 11196 -cneg 11394 -𝑒cxne 13038 |
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 5260 ax-nul 5267 ax-pow 5324 ax-un 7676 ax-cnex 11115 |
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 2941 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-uni 4870 df-iota 6452 df-fv 6508 df-ov 7364 df-pnf 11199 df-mnf 11200 df-xr 11201 df-neg 11396 df-xneg 13041 |
This theorem is referenced by: xrhmeo 24332 supminfxrrnmpt 43796 monoord2xrv 43809 liminfvalxr 44114 liminfpnfuz 44147 xlimpnfxnegmnf2 44189 |
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