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Theorem rpnegap 9632
Description: Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.)
Assertion
Ref Expression
rpnegap  |-  ( ( A  e.  RR  /\  A #  0 )  ->  ( A  e.  RR+  \/_  -u A  e.  RR+ ) )

Proof of Theorem rpnegap
StepHypRef Expression
1 0re 7909 . . . . . . 7  |-  0  e.  RR
2 reapltxor 8497 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A #  0  <->  ( A  <  0  \/_  0  <  A ) ) )
31, 2mpan2 423 . . . . . 6  |-  ( A  e.  RR  ->  ( A #  0  <->  ( A  <  0  \/_  0  < 
A ) ) )
4 xorcom 1383 . . . . . 6  |-  ( ( A  <  0  \/_  0  <  A )  <-> 
( 0  <  A  \/_  A  <  0 ) )
53, 4bitrdi 195 . . . . 5  |-  ( A  e.  RR  ->  ( A #  0  <->  ( 0  < 
A  \/_  A  <  0 ) ) )
65pm5.32i 451 . . . 4  |-  ( ( A  e.  RR  /\  A #  0 )  <->  ( A  e.  RR  /\  ( 0  <  A  \/_  A  <  0 ) ) )
7 anxordi 1395 . . . 4  |-  ( ( A  e.  RR  /\  ( 0  <  A  \/_  A  <  0 ) )  <->  ( ( A  e.  RR  /\  0  <  A )  \/_  ( A  e.  RR  /\  A  <  0 ) ) )
86, 7bitri 183 . . 3  |-  ( ( A  e.  RR  /\  A #  0 )  <->  ( ( A  e.  RR  /\  0  <  A )  \/_  ( A  e.  RR  /\  A  <  0 ) ) )
98biimpi 119 . 2  |-  ( ( A  e.  RR  /\  A #  0 )  ->  (
( A  e.  RR  /\  0  <  A ) 
\/_  ( A  e.  RR  /\  A  <  0 ) ) )
10 elrp 9601 . . . 4  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
1110a1i 9 . . 3  |-  ( ( A  e.  RR  /\  A #  0 )  ->  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) ) )
12 elrp 9601 . . . . . 6  |-  ( -u A  e.  RR+  <->  ( -u A  e.  RR  /\  0  <  -u A ) )
13 renegcl 8169 . . . . . . 7  |-  ( A  e.  RR  ->  -u A  e.  RR )
1413biantrurd 303 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <  -u A  <->  ( -u A  e.  RR  /\  0  <  -u A ) ) )
1512, 14bitr4id 198 . . . . 5  |-  ( A  e.  RR  ->  ( -u A  e.  RR+  <->  0  <  -u A ) )
16 lt0neg1 8376 . . . . 5  |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <  -u A ) )
17 ibar 299 . . . . 5  |-  ( A  e.  RR  ->  ( A  <  0  <->  ( A  e.  RR  /\  A  <  0 ) ) )
1815, 16, 173bitr2d 215 . . . 4  |-  ( A  e.  RR  ->  ( -u A  e.  RR+  <->  ( A  e.  RR  /\  A  <  0 ) ) )
1918adantr 274 . . 3  |-  ( ( A  e.  RR  /\  A #  0 )  ->  ( -u A  e.  RR+  <->  ( A  e.  RR  /\  A  <  0 ) ) )
2011, 19xorbi12d 1377 . 2  |-  ( ( A  e.  RR  /\  A #  0 )  ->  (
( A  e.  RR+  \/_  -u A  e.  RR+ )  <->  ( ( A  e.  RR  /\  0  <  A ) 
\/_  ( A  e.  RR  /\  A  <  0 ) ) ) )
219, 20mpbird 166 1  |-  ( ( A  e.  RR  /\  A #  0 )  ->  ( A  e.  RR+  \/_  -u A  e.  RR+ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/_ wxo 1370    e. wcel 2141   class class class wbr 3987   RRcr 7762   0cc0 7763    < clt 7943   -ucneg 8080   # cap 8489   RR+crp 9599
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-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-xor 1371  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-nel 2436  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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7945  df-mnf 7946  df-ltxr 7948  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-rp 9600
This theorem is referenced by: (None)
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