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Theorem xnn0nnn0pnf 8682
Description: An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
xnn0nnn0pnf  |-  ( ( N  e. NN0*  /\  -.  N  e.  NN0 )  ->  N  = +oo )

Proof of Theorem xnn0nnn0pnf
StepHypRef Expression
1 elxnn0 8671 . . 3  |-  ( N  e. NN0* 
<->  ( N  e.  NN0  \/  N  = +oo )
)
2 pm2.53 674 . . 3  |-  ( ( N  e.  NN0  \/  N  = +oo )  ->  ( -.  N  e. 
NN0  ->  N  = +oo ) )
31, 2sylbi 119 . 2  |-  ( N  e. NN0*  ->  ( -.  N  e.  NN0  ->  N  = +oo ) )
43imp 122 1  |-  ( ( N  e. NN0*  /\  -.  N  e.  NN0 )  ->  N  = +oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662    = wceq 1287    e. wcel 1436   +oocpnf 7463   NN0cn0 8606  NN0*cxnn0 8669
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-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 3932  ax-pow 3984  ax-un 4234  ax-cnex 7380
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-uni 3637  df-pnf 7468  df-xr 7470  df-xnn0 8670
This theorem is referenced by: (None)
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