ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xnn0xrnemnf Unicode version

Theorem xnn0xrnemnf 8674
Description: The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
xnn0xrnemnf  |-  ( A  e. NN0*  ->  ( A  e. 
RR*  /\  A  =/= -oo ) )

Proof of Theorem xnn0xrnemnf
StepHypRef Expression
1 xnn0xr 8667 . 2  |-  ( A  e. NN0*  ->  A  e.  RR* )
2 xnn0nemnf 8673 . 2  |-  ( A  e. NN0*  ->  A  =/= -oo )
31, 2jca 300 1  |-  ( A  e. NN0*  ->  ( A  e. 
RR*  /\  A  =/= -oo ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1436    =/= wne 2251   -oocmnf 7457   RR*cxr 7458  NN0*cxnn0 8662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3931  ax-pow 3983  ax-un 4233  ax-setind 4325  ax-cnex 7373  ax-resscn 7374  ax-1re 7376  ax-addrcl 7379  ax-rnegex 7391
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-uni 3637  df-int 3672  df-pnf 7461  df-mnf 7462  df-xr 7463  df-inn 8351  df-n0 8600  df-xnn0 8663
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator