Theorem xnn0nnn0pnf 9046

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 9035	. . 3 |- ( N e. NN0* <-> ( N e. NN0 \/ N = +oo ) )
2		pm2.53 711	. . 3 |- ( ( N e. NN0 \/ N = +oo ) -> ( -. N e. NN0 -> N = +oo ) )
3	1, 2	sylbi 120	. 2 |- ( N e. NN0* -> ( -. N e. NN0 -> N = +oo ) )
4	3	imp 123	1 |- ( ( N e. NN0* /\ -. N e. NN0 ) -> N = +oo )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   +oocpnf 7790   NN0cn0 8970  NN0*cxnn0 9033

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 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-un 4350  ax-cnex 7704

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-pnf 7795  df-xr 7797  df-xnn0 9034

This theorem is referenced by: (None)