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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 7189 | . . 3 | |
2 | 1 | adantr 261 | . 2 |
3 | simpl 102 | . . . 4 | |
4 | simpr 103 | . . . 4 | |
5 | addcl 7006 | . . . 4 | |
6 | 3, 4, 5 | syl2anr 274 | . . 3 |
7 | simplrr 488 | . . . . . . . 8 | |
8 | 7 | oveq1d 5527 | . . . . . . 7 |
9 | simplll 485 | . . . . . . . 8 | |
10 | simplrl 487 | . . . . . . . 8 | |
11 | simpllr 486 | . . . . . . . 8 | |
12 | 9, 10, 11 | addassd 7049 | . . . . . . 7 |
13 | 11 | addid2d 7163 | . . . . . . 7 |
14 | 8, 12, 13 | 3eqtr3rd 2081 | . . . . . 6 |
15 | 14 | eqeq2d 2051 | . . . . 5 |
16 | simpr 103 | . . . . . 6 | |
17 | 10, 11 | addcld 7046 | . . . . . 6 |
18 | 9, 16, 17 | addcand 7195 | . . . . 5 |
19 | 15, 18 | bitrd 177 | . . . 4 |
20 | 19 | ralrimiva 2392 | . . 3 |
21 | reu6i 2732 | . . 3 | |
22 | 6, 20, 21 | syl2anc 391 | . 2 |
23 | 2, 22 | rexlimddv 2437 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 wrex 2307 wreu 2308 (class class class)co 5512 cc 6887 cc0 6889 caddc 6892 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: subval 7203 subcl 7210 subadd 7214 |
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