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Mirrors > Home > ILE Home > Th. List > negeu | Unicode version |
Description: Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negeu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnegex 7286 | . . 3 | |
2 | 1 | adantr 270 | . 2 |
3 | simpl 107 | . . . 4 | |
4 | simpr 108 | . . . 4 | |
5 | addcl 7098 | . . . 4 | |
6 | 3, 4, 5 | syl2anr 284 | . . 3 |
7 | simplrr 502 | . . . . . . . 8 | |
8 | 7 | oveq1d 5547 | . . . . . . 7 |
9 | simplll 499 | . . . . . . . 8 | |
10 | simplrl 501 | . . . . . . . 8 | |
11 | simpllr 500 | . . . . . . . 8 | |
12 | 9, 10, 11 | addassd 7141 | . . . . . . 7 |
13 | 11 | addid2d 7258 | . . . . . . 7 |
14 | 8, 12, 13 | 3eqtr3rd 2122 | . . . . . 6 |
15 | 14 | eqeq2d 2092 | . . . . 5 |
16 | simpr 108 | . . . . . 6 | |
17 | 10, 11 | addcld 7138 | . . . . . 6 |
18 | 9, 16, 17 | addcand 7292 | . . . . 5 |
19 | 15, 18 | bitrd 186 | . . . 4 |
20 | 19 | ralrimiva 2434 | . . 3 |
21 | reu6i 2783 | . . 3 | |
22 | 6, 20, 21 | syl2anc 403 | . 2 |
23 | 2, 22 | rexlimddv 2481 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 wrex 2349 wreu 2350 (class class class)co 5532 cc 6979 cc0 6981 caddc 6984 |
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-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-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 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-nf 1390 df-sb 1686 df-eu 1944 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: subval 7300 subcl 7307 subadd 7311 |
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