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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnegex 9782 | . . 3 | |
2 | 1 | adantr 465 | . 2 |
3 | simpl 457 | . . . 4 | |
4 | simpr 461 | . . . 4 | |
5 | addcl 9595 | . . . 4 | |
6 | 3, 4, 5 | syl2anr 478 | . . 3 |
7 | simplrr 762 | . . . . . . . 8 | |
8 | 7 | oveq1d 6311 | . . . . . . 7 |
9 | simplll 759 | . . . . . . . 8 | |
10 | simplrl 761 | . . . . . . . 8 | |
11 | simpllr 760 | . . . . . . . 8 | |
12 | 9, 10, 11 | addassd 9639 | . . . . . . 7 |
13 | 11 | addid2d 9802 | . . . . . . 7 |
14 | 8, 12, 13 | 3eqtr3rd 2507 | . . . . . 6 |
15 | 14 | eqeq2d 2471 | . . . . 5 |
16 | simpr 461 | . . . . . 6 | |
17 | 10, 11 | addcld 9636 | . . . . . 6 |
18 | 9, 16, 17 | addcand 9804 | . . . . 5 |
19 | 15, 18 | bitrd 253 | . . . 4 |
20 | 19 | ralrimiva 2871 | . . 3 |
21 | reu6i 3290 | . . 3 | |
22 | 6, 20, 21 | syl2anc 661 | . 2 |
23 | 2, 22 | rexlimddv 2953 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 <-> wb 184
/\ wa 369 = wceq 1395 e. wcel 1818
A. wral 2807 E. wrex 2808 E! wreu 2809
(class class class)co 6296 cc 9511 0 cc0 9513 caddc 9516 |
This theorem is referenced by: subcl 9842 subadd 9846 addinv 25354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-op 4036 df-uni 4250 df-br 4453 df-opab 4511 df-mpt 4512 df-id 4800 df-po 4805 df-so 4806 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-ov 6299 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-pnf 9651 df-mnf 9652 df-ltxr 9654 |
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