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Theorem ltxr 10457
 Description: The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
ltxr

Proof of Theorem ltxr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 4028 . . . . 5
2 df-3an 936 . . . . . 6
32opabbii 4083 . . . . 5
41, 3brab2ga 4763 . . . 4
54a1i 10 . . 3
6 brun 4069 . . . 4
7 brxp 4720 . . . . . . 7
8 elun 3316 . . . . . . . . . . 11
9 orcom 376 . . . . . . . . . . 11
108, 9bitri 240 . . . . . . . . . 10
11 elsncg 3662 . . . . . . . . . . 11
1211orbi1d 683 . . . . . . . . . 10
1310, 12syl5bb 248 . . . . . . . . 9
14 elsncg 3662 . . . . . . . . 9
1513, 14bi2anan9 843 . . . . . . . 8
16 andir 838 . . . . . . . 8
1715, 16syl6bb 252 . . . . . . 7
187, 17syl5bb 248 . . . . . 6
19 brxp 4720 . . . . . . 7
2011anbi1d 685 . . . . . . . 8
2120adantr 451 . . . . . . 7
2219, 21syl5bb 248 . . . . . 6
2318, 22orbi12d 690 . . . . 5
24 orass 510 . . . . 5
2523, 24syl6bb 252 . . . 4
266, 25syl5bb 248 . . 3
275, 26orbi12d 690 . 2
28 df-ltxr 8872 . . . 4
2928breqi 4029 . . 3
30 brun 4069 . . 3
3129, 30bitri 240 . 2
32 orass 510 . 2
3327, 31, 323bitr4g 279 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wo 357   wa 358   w3a 934   wceq 1623   wcel 1684   cun 3150  csn 3640   class class class wbr 4023  copab 4076   cxp 4687  cr 8736   cltrr 8741   cpnf 8864   cmnf 8865  cxr 8866   clt 8867 This theorem is referenced by:  xrltnr  10462  ltpnf  10463  mnflt  10464  mnfltpnf  10465  pnfnlt  10467  nltmnf  10468 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-ltxr 8872
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