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Mirrors > Home > ILE Home > Th. List > xnegcl | Unicode version |
Description: Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9706 | . 2 | |
2 | rexneg 9760 | . . . . 5 | |
3 | renegcl 8153 | . . . . 5 | |
4 | 2, 3 | eqeltrd 2241 | . . . 4 |
5 | 4 | rexrd 7942 | . . 3 |
6 | xnegeq 9757 | . . . 4 | |
7 | xnegpnf 9758 | . . . . 5 | |
8 | mnfxr 7949 | . . . . 5 | |
9 | 7, 8 | eqeltri 2237 | . . . 4 |
10 | 6, 9 | eqeltrdi 2255 | . . 3 |
11 | xnegeq 9757 | . . . 4 | |
12 | xnegmnf 9759 | . . . . 5 | |
13 | pnfxr 7945 | . . . . 5 | |
14 | 12, 13 | eqeltri 2237 | . . . 4 |
15 | 11, 14 | eqeltrdi 2255 | . . 3 |
16 | 5, 10, 15 | 3jaoi 1292 | . 2 |
17 | 1, 16 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3o 966 wceq 1342 wcel 2135 cr 7746 cpnf 7924 cmnf 7925 cxr 7926 cneg 8064 cxne 9699 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-addcom 7847 ax-addass 7849 ax-distr 7851 ax-i2m1 7852 ax-0id 7855 ax-rnegex 7856 ax-cnre 7858 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-if 3519 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-br 3980 df-opab 4041 df-id 4268 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-iota 5150 df-fun 5187 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-pnf 7929 df-mnf 7930 df-xr 7931 df-sub 8065 df-neg 8066 df-xneg 9702 |
This theorem is referenced by: xltneg 9766 xleneg 9767 xnegcld 9785 xnegdi 9798 xaddass2 9800 xleadd1 9805 xsubge0 9811 xrnegiso 11197 xrminmax 11200 xrmincl 11201 xrmin1inf 11202 xrmin2inf 11203 xrlemininf 11206 xrminltinf 11207 |
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