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Theorem xnn0nnn0pnf 9184
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 9173 . . 3  |-  ( N  e. NN0* 
<->  ( N  e.  NN0  \/  N  = +oo )
)
2 pm2.53 712 . . 3  |-  ( ( N  e.  NN0  \/  N  = +oo )  ->  ( -.  N  e. 
NN0  ->  N  = +oo ) )
31, 2sylbi 120 . 2  |-  ( N  e. NN0*  ->  ( -.  N  e.  NN0  ->  N  = +oo ) )
43imp 123 1  |-  ( ( N  e. NN0*  /\  -.  N  e.  NN0 )  ->  N  = +oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 698    = wceq 1342    e. wcel 2135   +oocpnf 7924   NN0cn0 9108  NN0*cxnn0 9171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-un 4408  ax-cnex 7838
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2726  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-uni 3787  df-pnf 7929  df-xr 7931  df-xnn0 9172
This theorem is referenced by: (None)
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