Theorem xnn0nemnf 9478

Description: No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)

Assertion

Ref	Expression
xnn0nemnf	|- ( A e. NN0* -> A =/= -oo )

Proof of Theorem xnn0nemnf

Step	Hyp	Ref	Expression
1		elxnn0 9469	. 2 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		nn0re 9413	. . . 4 |- ( A e. NN0 -> A e. RR )
3	2	renemnfd 8233	. . 3 |- ( A e. NN0 -> A =/= -oo )
4		pnfnemnf 8236	. . . 4 |- +oo =/= -oo
5		neeq1 2414	. . . 4 |- ( A = +oo -> ( A =/= -oo <-> +oo =/= -oo ) )
6	4, 5	mpbiri 168	. . 3 |- ( A = +oo -> A =/= -oo )
7	3, 6	jaoi 723	. 2 |- ( ( A e. NN0 \/ A = +oo ) -> A =/= -oo )
8	1, 7	sylbi 121	1 |- ( A e. NN0* -> A =/= -oo )

Syntax hints:   -> wi 4   \/ wo 715   = wceq 1397   e. wcel 2201   =/= wne 2401   +oocpnf 8213   -oocmnf 8214   NN0cn0 9404  NN0*cxnn0 9467

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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-pow 4263  ax-un 4529  ax-setind 4634  ax-cnex 8125  ax-resscn 8126  ax-1re 8128  ax-addrcl 8131  ax-rnegex 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-uni 3893  df-int 3928  df-pnf 8218  df-mnf 8219  df-xr 8220  df-inn 9146  df-n0 9405  df-xnn0 9468

This theorem is referenced by:  xnn0xrnemnf  9479