Theorem xnn0nnn0pnf 11376

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 11365	. . 3 |- ( N e. NN0* <-> ( N e. NN0 \/ N = +oo ) )
2		pm2.53 388	. . 3 |- ( ( N e. NN0 \/ N = +oo ) -> ( -. N e. NN0 -> N = +oo ) )
3	1, 2	sylbi 207	. 2 |- ( N e. NN0* -> ( -. N e. NN0 -> N = +oo ) )
4	3	imp 445	1 |- ( ( N e. NN0* /\ -. N e. NN0 ) -> N = +oo )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   +oocpnf 10071   NN0cn0 11292  NN0*cxnn0 11363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-pow 4843  ax-un 6949  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-pnf 10076  df-xr 10078  df-xnn0 11364

This theorem is referenced by:  xnn0xaddcl  12066