Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrge0nre Structured version   Unicode version

Theorem xrge0nre 27840
Description: An extended real which is not a real is plus infinity. (Contributed by Thierry Arnoux, 16-Oct-2017.)
Assertion
Ref Expression
xrge0nre  |-  ( ( A  e.  ( 0 [,] +oo )  /\  -.  A  e.  RR )  ->  A  = +oo )

Proof of Theorem xrge0nre
StepHypRef Expression
1 iccssxr 11632 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
21sseli 3495 . . 3  |-  ( A  e.  ( 0 [,] +oo )  ->  A  e. 
RR* )
3 xrge0neqmnf 27839 . . 3  |-  ( A  e.  ( 0 [,] +oo )  ->  A  =/= -oo )
4 xrnemnf 11353 . . . 4  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( A  e.  RR  \/  A  = +oo ) )
54biimpi 194 . . 3  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A  e.  RR  \/  A  = +oo )
)
62, 3, 5syl2anc 661 . 2  |-  ( A  e.  ( 0 [,] +oo )  ->  ( A  e.  RR  \/  A  = +oo ) )
76orcanai 913 1  |-  ( ( A  e.  ( 0 [,] +oo )  /\  -.  A  e.  RR )  ->  A  = +oo )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652  (class class class)co 6296   RRcr 9508   0cc0 9509   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644   [,]cicc 11557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-icc 11561
This theorem is referenced by:  voliune  28374  volfiniune  28375  omssubadd  28444
  Copyright terms: Public domain W3C validator