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Mirrors > Home > MPE 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnegex 9207 |
. . 3
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2 | 1 | adantr 452 |
. 2
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3 | simpl 444 |
. . . 4
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4 | simpr 448 |
. . . 4
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5 | addcl 9032 |
. . . 4
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6 | 3, 4, 5 | syl2anr 465 |
. . 3
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7 | simplrr 738 |
. . . . . . . 8
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8 | 7 | oveq1d 6059 |
. . . . . . 7
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9 | simplll 735 |
. . . . . . . 8
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10 | simplrl 737 |
. . . . . . . 8
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11 | simpllr 736 |
. . . . . . . 8
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12 | 9, 10, 11 | addassd 9070 |
. . . . . . 7
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13 | 11 | addid2d 9227 |
. . . . . . 7
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14 | 8, 12, 13 | 3eqtr3rd 2449 |
. . . . . 6
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15 | 14 | eqeq2d 2419 |
. . . . 5
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16 | simpr 448 |
. . . . . 6
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17 | 10, 11 | addcld 9067 |
. . . . . 6
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18 | 9, 16, 17 | addcand 9229 |
. . . . 5
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19 | 15, 18 | bitrd 245 |
. . . 4
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20 | 19 | ralrimiva 2753 |
. . 3
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21 | reu6i 3089 |
. . 3
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22 | 6, 20, 21 | syl2anc 643 |
. 2
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23 | 2, 22 | rexlimddv 2798 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: subcl 9265 subadd 9268 addinv 21897 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2389 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 ax-resscn 9007 ax-1cn 9008 ax-icn 9009 ax-addcl 9010 ax-addrcl 9011 ax-mulcl 9012 ax-mulrcl 9013 ax-mulcom 9014 ax-addass 9015 ax-mulass 9016 ax-distr 9017 ax-i2m1 9018 ax-1ne0 9019 ax-1rid 9020 ax-rnegex 9021 ax-rrecex 9022 ax-cnre 9023 ax-pre-lttri 9024 ax-pre-lttrn 9025 ax-pre-ltadd 9026 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-nel 2574 df-ral 2675 df-rex 2676 df-reu 2677 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-nul 3593 df-if 3704 df-pw 3765 df-sn 3784 df-pr 3785 df-op 3787 df-uni 3980 df-br 4177 df-opab 4231 df-mpt 4232 df-id 4462 df-po 4467 df-so 4468 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-res 4853 df-ima 4854 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-f1 5422 df-fo 5423 df-f1o 5424 df-fv 5425 df-ov 6047 df-er 6868 df-en 7073 df-dom 7074 df-sdom 7075 df-pnf 9082 df-mnf 9083 df-ltxr 9085 |
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