Theorem xnn0lenn0nn0 9901

Description: An extended nonnegative integer which is less than or equal to a nonnegative integer is a nonnegative integer. (Contributed by AV, 24-Nov-2021.)

Assertion

Ref	Expression
xnn0lenn0nn0	|- ( ( M e. NN0* /\ N e. NN0 /\ M <_ N ) -> M e. NN0 )

Proof of Theorem xnn0lenn0nn0

Step	Hyp	Ref	Expression
1		elxnn0 9276	. . 3 |- ( M e. NN0* <-> ( M e. NN0 \/ M = +oo ) )
2		2a1 25	. . . 4 |- ( M e. NN0 -> ( N e. NN0 -> ( M <_ N -> M e. NN0 ) ) )
3		breq1 4024	. . . . . . 7 |- ( M = +oo -> ( M <_ N <-> +oo <_ N ) )
4	3	adantr 276	. . . . . 6 |- ( ( M = +oo /\ N e. NN0 ) -> ( M <_ N <-> +oo <_ N ) )
5		nn0re 9220	. . . . . . . . . 10 |- ( N e. NN0 -> N e. RR )
6	5	rexrd 8042	. . . . . . . . 9 |- ( N e. NN0 -> N e. RR* )
7		xgepnf 9852	. . . . . . . . 9 |- ( N e. RR* -> ( +oo <_ N <-> N = +oo ) )
8	6, 7	syl 14	. . . . . . . 8 |- ( N e. NN0 -> ( +oo <_ N <-> N = +oo ) )
9		pnfnre 8034	. . . . . . . . 9 |- +oo e/ RR
10		eleq1 2252	. . . . . . . . . . 11 |- ( N = +oo -> ( N e. NN0 <-> +oo e. NN0 ) )
11		nn0re 9220	. . . . . . . . . . . 12 |- ( +oo e. NN0 -> +oo e. RR )
12		elnelall 2467	. . . . . . . . . . . 12 |- ( +oo e. RR -> ( +oo e/ RR -> M e. NN0 ) )
13	11, 12	syl 14	. . . . . . . . . . 11 |- ( +oo e. NN0 -> ( +oo e/ RR -> M e. NN0 ) )
14	10, 13	biimtrdi 163	. . . . . . . . . 10 |- ( N = +oo -> ( N e. NN0 -> ( +oo e/ RR -> M e. NN0 ) ) )
15	14	com13 80	. . . . . . . . 9 |- ( +oo e/ RR -> ( N e. NN0 -> ( N = +oo -> M e. NN0 ) ) )
16	9, 15	ax-mp 5	. . . . . . . 8 |- ( N e. NN0 -> ( N = +oo -> M e. NN0 ) )
17	8, 16	sylbid 150	. . . . . . 7 |- ( N e. NN0 -> ( +oo <_ N -> M e. NN0 ) )
18	17	adantl 277	. . . . . 6 |- ( ( M = +oo /\ N e. NN0 ) -> ( +oo <_ N -> M e. NN0 ) )
19	4, 18	sylbid 150	. . . . 5 |- ( ( M = +oo /\ N e. NN0 ) -> ( M <_ N -> M e. NN0 ) )
20	19	ex 115	. . . 4 |- ( M = +oo -> ( N e. NN0 -> ( M <_ N -> M e. NN0 ) ) )
21	2, 20	jaoi 717	. . 3 |- ( ( M e. NN0 \/ M = +oo ) -> ( N e. NN0 -> ( M <_ N -> M e. NN0 ) ) )
22	1, 21	sylbi 121	. 2 |- ( M e. NN0* -> ( N e. NN0 -> ( M <_ N -> M e. NN0 ) ) )
23	22	3imp 1195	1 |- ( ( M e. NN0* /\ N e. NN0 /\ M <_ N ) -> M e. NN0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2160   e/ wnel 2455   class class class wbr 4021   RRcr 7845   +oocpnf 8024   RR*cxr 8026   <_ cle 8028   NN0cn0 9211  NN0*cxnn0 9274

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1re 7940  ax-addrcl 7943  ax-rnegex 7955  ax-pre-ltirr 7958

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-xp 4653  df-cnv 4655  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-inn 8955  df-n0 9212  df-xnn0 9275

This theorem is referenced by:  xnn0le2is012  9902