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Theorem xrltnsym 11970
Description: Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
Assertion
Ref Expression
xrltnsym |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -. B < A ) )
Proof of Theorem xrltnsym
StepHypRef Expression
1 elxr 11950 . 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2 elxr 11950 . 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3 ltnsym 10135 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
4 rexr 10085 . . . . . . . 8 |- ( A e. RR -> A e. RR* )
5 pnfnlt 11962 . . . . . . . 8 |- ( A e. RR* -> -. +oo < A )
64, 5syl 17 . . . . . . 7 |- ( A e. RR -> -. +oo < A )
76adantr 481 . . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> -. +oo < A )
8 breq1 4656 . . . . . . 7 |- ( B = +oo -> ( B < A <-> +oo < A ) )
98adantl 482 . . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( B < A <-> +oo < A ) )
107, 9mtbird 315 . . . . 5 |- ( ( A e. RR /\ B = +oo ) -> -. B < A )
1110a1d 25 . . . 4 |- ( ( A e. RR /\ B = +oo ) -> ( A < B -> -. B < A ) )
12 nltmnf 11963 . . . . . . . 8 |- ( A e. RR* -> -. A < -oo )
134, 12syl 17 . . . . . . 7 |- ( A e. RR -> -. A < -oo )
1413adantr 481 . . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
15 breq2 4657 . . . . . . 7 |- ( B = -oo -> ( A < B <-> A < -oo ) )
1615adantl 482 . . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
1714, 16mtbird 315 . . . . 5 |- ( ( A e. RR /\ B = -oo ) -> -. A < B )
1817pm2.21d 118 . . . 4 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> -. B < A ) )
193, 11, 183jaodan 1394 . . 3 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
20 pnfnlt 11962 . . . . . . 7 |- ( B e. RR* -> -. +oo < B )
2120adantl 482 . . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
22 breq1 4656 . . . . . . 7 |- ( A = +oo -> ( A < B <-> +oo < B ) )
2322adantr 481 . . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
2421, 23mtbird 315 . . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> -. A < B )
2524pm2.21d 118 . . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> -. B < A ) )
262, 25sylan2br 493 . . 3 |- ( ( A = +oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
27 rexr 10085 . . . . . . . 8 |- ( B e. RR -> B e. RR* )
28 nltmnf 11963 . . . . . . . 8 |- ( B e. RR* -> -. B < -oo )
2927, 28syl 17 . . . . . . 7 |- ( B e. RR -> -. B < -oo )
3029adantl 482 . . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> -. B < -oo )
31 breq2 4657 . . . . . . 7 |- ( A = -oo -> ( B < A <-> B < -oo ) )
3231adantr 481 . . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> ( B < A <-> B < -oo ) )
3330, 32mtbird 315 . . . . 5 |- ( ( A = -oo /\ B e. RR ) -> -. B < A )
3433a1d 25 . . . 4 |- ( ( A = -oo /\ B e. RR ) -> ( A < B -> -. B < A ) )
35 mnfxr 10096 . . . . . . . 8 |- -oo e. RR*
36 pnfnlt 11962 . . . . . . . 8 |- ( -oo e. RR* -> -. +oo < -oo )
3735, 36ax-mp 5 . . . . . . 7 |- -. +oo < -oo
38 breq12 4658 . . . . . . 7 |- ( ( B = +oo /\ A = -oo ) -> ( B < A <-> +oo < -oo ) )
3937, 38mtbiri 317 . . . . . 6 |- ( ( B = +oo /\ A = -oo ) -> -. B < A )
4039ancoms 469 . . . . 5 |- ( ( A = -oo /\ B = +oo ) -> -. B < A )
4140a1d 25 . . . 4 |- ( ( A = -oo /\ B = +oo ) -> ( A < B -> -. B < A ) )
42 xrltnr 11953 . . . . . . 7 |- ( -oo e. RR* -> -. -oo < -oo )
4335, 42ax-mp 5 . . . . . 6 |- -. -oo < -oo
44 breq12 4658 . . . . . 6 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
4543, 44mtbiri 317 . . . . 5 |- ( ( A = -oo /\ B = -oo ) -> -. A < B )
4645pm2.21d 118 . . . 4 |- ( ( A = -oo /\ B = -oo ) -> ( A < B -> -. B < A ) )
4734, 41, 463jaodan 1394 . . 3 |- ( ( A = -oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
4819, 26, 473jaoian 1393 . 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> -. B < A ) )
491, 2, 48syl2anb 496 1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -. B < A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   class class class wbr 4653   RRcr 9935   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079
This theorem is referenced by:  xrltnsym2  11971  xrlttri  11972  xmullem2  12095  sgnp  13830  iccpartnel  41374
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