Theorem xnn0nemnf 9443

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 9434	. 2 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		nn0re 9378	. . . 4 |- ( A e. NN0 -> A e. RR )
3	2	renemnfd 8198	. . 3 |- ( A e. NN0 -> A =/= -oo )
4		pnfnemnf 8201	. . . 4 |- +oo =/= -oo
5		neeq1 2413	. . . 4 |- ( A = +oo -> ( A =/= -oo <-> +oo =/= -oo ) )
6	4, 5	mpbiri 168	. . 3 |- ( A = +oo -> A =/= -oo )
7	3, 6	jaoi 721	. 2 |- ( ( A e. NN0 \/ A = +oo ) -> A =/= -oo )
8	1, 7	sylbi 121	1 |- ( A e. NN0* -> A =/= -oo )

Syntax hints:   -> wi 4   \/ wo 713   = wceq 1395   e. wcel 2200   =/= wne 2400   +oocpnf 8178   -oocmnf 8179   NN0cn0 9369  NN0*cxnn0 9432

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096  ax-rnegex 8108

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-int 3924  df-pnf 8183  df-mnf 8184  df-xr 8185  df-inn 9111  df-n0 9370  df-xnn0 9433

This theorem is referenced by:  xnn0xrnemnf  9444