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Theorem rpnegap 8836
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 7170 . . . . . . 7  |-  0  e.  RR
2 reapltxor 7745 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A #  0  <->  ( A  <  0  \/_  0  <  A ) ) )
31, 2mpan2 416 . . . . . 6  |-  ( A  e.  RR  ->  ( A #  0  <->  ( A  <  0  \/_  0  < 
A ) ) )
4 xorcom 1320 . . . . . 6  |-  ( ( A  <  0  \/_  0  <  A )  <-> 
( 0  <  A  \/_  A  <  0 ) )
53, 4syl6bb 194 . . . . 5  |-  ( A  e.  RR  ->  ( A #  0  <->  ( 0  < 
A  \/_  A  <  0 ) ) )
65pm5.32i 442 . . . 4  |-  ( ( A  e.  RR  /\  A #  0 )  <->  ( A  e.  RR  /\  ( 0  <  A  \/_  A  <  0 ) ) )
7 anxordi 1332 . . . 4  |-  ( ( A  e.  RR  /\  ( 0  <  A  \/_  A  <  0 ) )  <->  ( ( A  e.  RR  /\  0  <  A )  \/_  ( A  e.  RR  /\  A  <  0 ) ) )
86, 7bitri 182 . . 3  |-  ( ( A  e.  RR  /\  A #  0 )  <->  ( ( A  e.  RR  /\  0  <  A )  \/_  ( A  e.  RR  /\  A  <  0 ) ) )
98biimpi 118 . 2  |-  ( ( A  e.  RR  /\  A #  0 )  ->  (
( A  e.  RR  /\  0  <  A ) 
\/_  ( A  e.  RR  /\  A  <  0 ) ) )
10 elrp 8806 . . . 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 renegcl 7425 . . . . . . 7  |-  ( A  e.  RR  ->  -u A  e.  RR )
1312biantrurd 299 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <  -u A  <->  ( -u A  e.  RR  /\  0  <  -u A ) ) )
14 elrp 8806 . . . . . 6  |-  ( -u A  e.  RR+  <->  ( -u A  e.  RR  /\  0  <  -u A ) )
1513, 14syl6rbbr 197 . . . . 5  |-  ( A  e.  RR  ->  ( -u A  e.  RR+  <->  0  <  -u A ) )
16 lt0neg1 7628 . . . . 5  |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <  -u A ) )
17 ibar 295 . . . . 5  |-  ( A  e.  RR  ->  ( A  <  0  <->  ( A  e.  RR  /\  A  <  0 ) ) )
1815, 16, 173bitr2d 214 . . . 4  |-  ( A  e.  RR  ->  ( -u A  e.  RR+  <->  ( A  e.  RR  /\  A  <  0 ) ) )
1918adantr 270 . . 3  |-  ( ( A  e.  RR  /\  A #  0 )  ->  ( -u A  e.  RR+  <->  ( A  e.  RR  /\  A  <  0 ) ) )
2011, 19xorbi12d 1314 . 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 165 1  |-  ( ( A  e.  RR  /\  A #  0 )  ->  ( A  e.  RR+  \/_  -u A  e.  RR+ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/_ wxo 1307    e. wcel 1434   class class class wbr 3787   RRcr 7031   0cc0 7032    < clt 7204   -ucneg 7336   # cap 7737   RR+crp 8804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-mulrcl 7126  ax-addcom 7127  ax-mulcom 7128  ax-addass 7129  ax-mulass 7130  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-1rid 7134  ax-0id 7135  ax-rnegex 7136  ax-precex 7137  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143  ax-pre-mulgt0 7144
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-xor 1308  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-po 4053  df-iso 4054  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-pnf 7206  df-mnf 7207  df-ltxr 7209  df-sub 7337  df-neg 7338  df-reap 7731  df-ap 7738  df-rp 8805
This theorem is referenced by: (None)
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