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Theorem ltrelxr 9669
 Description: 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
ltrelxr

Proof of Theorem ltrelxr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltxr 9654 . 2
2 df-3an 975 . . . . . 6
32opabbii 4516 . . . . 5
4 opabssxp 5079 . . . . 5
53, 4eqsstri 3533 . . . 4
6 rexpssxrxp 9659 . . . 4
75, 6sstri 3512 . . 3
8 ressxr 9658 . . . . . 6
9 snsspr2 4180 . . . . . . 7
10 ssun2 3667 . . . . . . . 8
11 df-xr 9653 . . . . . . . 8
1210, 11sseqtr4i 3536 . . . . . . 7
139, 12sstri 3512 . . . . . 6
148, 13unssi 3678 . . . . 5
15 snsspr1 4179 . . . . . 6
1615, 12sstri 3512 . . . . 5
17 xpss12 5113 . . . . 5
1814, 16, 17mp2an 672 . . . 4
19 xpss12 5113 . . . . 5
2013, 8, 19mp2an 672 . . . 4
2118, 20unssi 3678 . . 3
227, 21unssi 3678 . 2
231, 22eqsstri 3533 1
 Colors of variables: wff setvar class Syntax hints:  /\wa 369  /\w3a 973  e.wcel 1818  u.cun 3473  C_wss 3475  {csn 4029  {cpr 4031   class class class wbr 4452  {copab 4509  X.cxp 5002   cr 9512   cltrr 9517   cpnf 9646   cmnf 9647   cxr 9648   clt 9649 This theorem is referenced by:  ltrel  9670  dfle2  11382  dflt2  11383  itg2gt0cn  30070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-in 3482  df-ss 3489  df-pr 4032  df-opab 4511  df-xp 5010  df-xr 9653  df-ltxr 9654
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