Theorem xnn0nnn0pnf 9319

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

Step	Hyp	Ref	Expression
1		elxnn0 9308	. . 3 |- ( N e. NN0* <-> ( N e. NN0 \/ N = +oo ) )
2		pm2.53 723	. . 3 |- ( ( N e. NN0 \/ N = +oo ) -> ( -. N e. NN0 -> N = +oo ) )
3	1, 2	sylbi 121	. 2 |- ( N e. NN0* -> ( -. N e. NN0 -> N = +oo ) )
4	3	imp 124	1 |- ( ( N e. NN0* /\ -. N e. NN0 ) -> N = +oo )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2164   +oocpnf 8053   NN0cn0 9243  NN0*cxnn0 9306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-un 4465  ax-cnex 7965

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-pnf 8058  df-xr 8060  df-xnn0 9307

This theorem is referenced by: (None)