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Mirrors > Home > ILE Home > Th. List > rpnegap | Unicode version |
Description: Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.) |
Ref | Expression |
---|---|
rpnegap | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7909 | . . . . . . 7 | |
2 | reapltxor 8497 | . . . . . . 7 # | |
3 | 1, 2 | mpan2 423 | . . . . . 6 # |
4 | xorcom 1383 | . . . . . 6 | |
5 | 3, 4 | bitrdi 195 | . . . . 5 # |
6 | 5 | pm5.32i 451 | . . . 4 # |
7 | anxordi 1395 | . . . 4 | |
8 | 6, 7 | bitri 183 | . . 3 # |
9 | 8 | biimpi 119 | . 2 # |
10 | elrp 9601 | . . . 4 | |
11 | 10 | a1i 9 | . . 3 # |
12 | elrp 9601 | . . . . . 6 | |
13 | renegcl 8169 | . . . . . . 7 | |
14 | 13 | biantrurd 303 | . . . . . 6 |
15 | 12, 14 | bitr4id 198 | . . . . 5 |
16 | lt0neg1 8376 | . . . . 5 | |
17 | ibar 299 | . . . . 5 | |
18 | 15, 16, 17 | 3bitr2d 215 | . . . 4 |
19 | 18 | adantr 274 | . . 3 # |
20 | 11, 19 | xorbi12d 1377 | . 2 # |
21 | 9, 20 | mpbird 166 | 1 # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wxo 1370 wcel 2141 class class class wbr 3987 cr 7762 cc0 7763 clt 7943 cneg 8080 # cap 8489 crp 9599 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-xor 1371 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7945 df-mnf 7946 df-ltxr 7948 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 df-rp 9600 |
This theorem is referenced by: (None) |
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