Theorem xnn0nnn0pnf 9211

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 9200	. . 3 |- ( N e. NN0* <-> ( N e. NN0 \/ N = +oo ) )
2		pm2.53 717	. . 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 703   = wceq 1348   e. wcel 2141   +oocpnf 7951   NN0cn0 9135  NN0*cxnn0 9198

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 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-un 4418  ax-cnex 7865

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-pnf 7956  df-xr 7958  df-xnn0 9199

This theorem is referenced by: (None)