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Mirrors > Home > ILE Home > Th. List > negm | Unicode version |
Description: The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.) |
Ref | Expression |
---|---|
negm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2993 | . . . . 5 | |
2 | renegcl 7369 | . . . . . . . 8 | |
3 | negeq 7301 | . . . . . . . . . 10 | |
4 | 3 | eleq1d 2147 | . . . . . . . . 9 |
5 | 4 | elrab3 2750 | . . . . . . . 8 |
6 | 2, 5 | syl 14 | . . . . . . 7 |
7 | recn 7106 | . . . . . . . . 9 | |
8 | 7 | negnegd 7410 | . . . . . . . 8 |
9 | 8 | eleq1d 2147 | . . . . . . 7 |
10 | 6, 9 | bitrd 186 | . . . . . 6 |
11 | 10 | biimprd 156 | . . . . 5 |
12 | 1, 11 | syli 37 | . . . 4 |
13 | elex2 2615 | . . . 4 | |
14 | 12, 13 | syl6 33 | . . 3 |
15 | 14 | exlimdv 1740 | . 2 |
16 | 15 | imp 122 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 crab 2352 wss 2973 cr 6980 cneg 7280 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-neg 7282 |
This theorem is referenced by: (None) |
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