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Theorem neg11 8170
Description: Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
neg11  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  = 
-u B  <->  A  =  B ) )

Proof of Theorem neg11
StepHypRef Expression
1 negeq 8112 . . 3  |-  ( -u A  =  -u B  ->  -u -u A  =  -u -u B
)
2 negneg 8169 . . . 4  |-  ( A  e.  CC  ->  -u -u A  =  A )
3 negneg 8169 . . . 4  |-  ( B  e.  CC  ->  -u -u B  =  B )
42, 3eqeqan12d 2186 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u -u A  =  -u -u B  <->  A  =  B ) )
51, 4syl5ib 153 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  = 
-u B  ->  A  =  B ) )
6 negeq 8112 . 2  |-  ( A  =  B  ->  -u A  =  -u B )
75, 6impbid1 141 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  = 
-u B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   CCcc 7772   -ucneg 8091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-neg 8093
This theorem is referenced by:  negcon1  8171  negeq0  8173  neg11i  8200  neg11ad  8226  subeqrev  8295
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