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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 12495 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
2 | mnfxr 10686 | . . . 4 ⊢ -∞ ∈ ℝ* | |
3 | 2 | elexi 3511 | . . 3 ⊢ -∞ ∈ V |
4 | pnfex 10682 | . . . 4 ⊢ +∞ ∈ V | |
5 | negex 10872 | . . . 4 ⊢ -𝐴 ∈ V | |
6 | 4, 5 | ifex 4511 | . . 3 ⊢ if(𝐴 = -∞, +∞, -𝐴) ∈ V |
7 | 3, 6 | ifex 4511 | . 2 ⊢ if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) ∈ V |
8 | 1, 7 | eqeltri 2906 | 1 ⊢ -𝑒𝐴 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ifcif 4463 +∞cpnf 10660 -∞cmnf 10661 ℝ*cxr 10662 -cneg 10859 -𝑒cxne 12492 |
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-un 7450 ax-cnex 10581 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 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-ral 3140 df-rex 3141 df-v 3494 df-sbc 3770 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-uni 4831 df-iota 6307 df-fv 6356 df-ov 7148 df-pnf 10665 df-mnf 10666 df-xr 10667 df-neg 10861 df-xneg 12495 |
This theorem is referenced by: xrhmeo 23477 supminfxrrnmpt 41623 monoord2xrv 41636 liminfvalxr 41940 liminfpnfuz 41973 xlimpnfxnegmnf2 42015 |
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