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Theorem xrge0nre 26158
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 11383 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
21sseli 3357 . . 3  |-  ( A  e.  ( 0 [,] +oo )  ->  A  e. 
RR* )
3 xrge0neqmnf 26157 . . 3  |-  ( A  e.  ( 0 [,] +oo )  ->  A  =/= -oo )
4 xrnemnf 11104 . . . 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 904 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 1369    e. wcel 1756    =/= wne 2611  (class class class)co 6096   RRcr 9286   0cc0 9287   +oocpnf 9420   -oocmnf 9421   RR*cxr 9422   [,]cicc 11308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-i2m1 9355  ax-1ne0 9356  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-icc 11312
This theorem is referenced by:  voliune  26650  volfiniune  26651
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