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Theorem xrltnsym 8484
Description: Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
Assertion
Ref Expression
xrltnsym  RR*  RR*  <  <

Proof of Theorem xrltnsym
StepHypRef Expression
1 elxr 8466 . 2  RR*  RR +oo -oo
2 elxr 8466 . 2  RR*  RR +oo -oo
3 ltnsym 6901 . . . 4  RR  RR  <  <
4 rexr 6868 . . . . . . . 8  RR  RR*
5 pnfnlt 8478 . . . . . . . 8  RR* +oo  <
64, 5syl 14 . . . . . . 7  RR +oo 
<
76adantr 261 . . . . . 6  RR +oo +oo  <
8 breq1 3758 . . . . . . 7 +oo  < +oo  <
98adantl 262 . . . . . 6  RR +oo  < +oo  <
107, 9mtbird 597 . . . . 5  RR +oo  <
1110a1d 22 . . . 4  RR +oo  <  <
12 nltmnf 8479 . . . . . . . 8  RR*  < -oo
134, 12syl 14 . . . . . . 7  RR  < -oo
1413adantr 261 . . . . . 6  RR -oo  < -oo
15 breq2 3759 . . . . . . 7 -oo  <  < -oo
1615adantl 262 . . . . . 6  RR -oo  <  < -oo
1714, 16mtbird 597 . . . . 5  RR -oo  <
1817pm2.21d 549 . . . 4  RR -oo  <  <
193, 11, 183jaodan 1200 . . 3  RR  RR +oo -oo  <  <
20 pnfnlt 8478 . . . . . . 7  RR* +oo  <
2120adantl 262 . . . . . 6 +oo  RR* +oo  <
22 breq1 3758 . . . . . . 7 +oo  < +oo  <
2322adantr 261 . . . . . 6 +oo  RR*  < +oo  <
2421, 23mtbird 597 . . . . 5 +oo  RR*  <
2524pm2.21d 549 . . . 4 +oo  RR*  <  <
262, 25sylan2br 272 . . 3 +oo  RR +oo -oo  <  <
27 rexr 6868 . . . . . . . 8  RR  RR*
28 nltmnf 8479 . . . . . . . 8  RR*  < -oo
2927, 28syl 14 . . . . . . 7  RR  < -oo
3029adantl 262 . . . . . 6 -oo  RR  < -oo
31 breq2 3759 . . . . . . 7 -oo  <  < -oo
3231adantr 261 . . . . . 6 -oo  RR  <  < -oo
3330, 32mtbird 597 . . . . 5 -oo  RR  <
3433a1d 22 . . . 4 -oo  RR  <  <
35 mnfxr 8464 . . . . . . . 8 -oo  RR*
36 pnfnlt 8478 . . . . . . . 8 -oo  RR* +oo  < -oo
3735, 36ax-mp 7 . . . . . . 7 +oo  < -oo
38 breq12 3760 . . . . . . 7 +oo -oo  < +oo  < -oo
3937, 38mtbiri 599 . . . . . 6 +oo -oo  <
4039ancoms 255 . . . . 5 -oo +oo  <
4140a1d 22 . . . 4 -oo +oo  <  <
42 xrltnr 8471 . . . . . . 7 -oo  RR* -oo  < -oo
4335, 42ax-mp 7 . . . . . 6 -oo  < -oo
44 breq12 3760 . . . . . 6 -oo -oo  < -oo  < -oo
4543, 44mtbiri 599 . . . . 5 -oo -oo  <
4645pm2.21d 549 . . . 4 -oo -oo  <  <
4734, 41, 463jaodan 1200 . . 3 -oo  RR +oo -oo  <  <
4819, 26, 473jaoian 1199 . 2  RR +oo -oo  RR +oo -oo  <  <
491, 2, 48syl2anb 275 1  RR*  RR*  <  <
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   w3o 883   wceq 1242   wcel 1390   class class class wbr 3755   RRcr 6710   +oocpnf 6854   -oocmnf 6855   RR*cxr 6856    < clt 6857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-pre-ltirr 6795  ax-pre-lttrn 6797
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862
This theorem is referenced by:  xrltnsym2  8485  xrltle  8489
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