Theorem xnn0lenn0nn0 10198

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 9565	. . 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 4112	. . . . . . 7 |- ( M = +oo -> ( M <_ N <-> +oo <_ N ) )
4	3	adantr 276	. . . . . 6 |- ( ( M = +oo /\ N e. NN0 ) -> ( M <_ N <-> +oo <_ N ) )
5		nn0re 9505	. . . . . . . . . 10 |- ( N e. NN0 -> N e. RR )
6	5	rexrd 8323	. . . . . . . . 9 |- ( N e. NN0 -> N e. RR* )
7		xgepnf 10149	. . . . . . . . 9 |- ( N e. RR* -> ( +oo <_ N <-> N = +oo ) )
8	6, 7	syl 14	. . . . . . . 8 |- ( N e. NN0 -> ( +oo <_ N <-> N = +oo ) )
9		pnfnre 8315	. . . . . . . . 9 |- +oo e/ RR
10		eleq1 2295	. . . . . . . . . . 11 |- ( N = +oo -> ( N e. NN0 <-> +oo e. NN0 ) )
11		nn0re 9505	. . . . . . . . . . . 12 |- ( +oo e. NN0 -> +oo e. RR )
12		elnelall 2519	. . . . . . . . . . . 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 724	. . 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 1220	1 |- ( ( M e. NN0* /\ N e. NN0 /\ M <_ N ) -> M e. NN0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   e/ wnel 2507   class class class wbr 4109   RRcr 8126   +oocpnf 8305   RR*cxr 8307   <_ cle 8309   NN0cn0 9496  NN0*cxnn0 9563

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224  ax-rnegex 8236  ax-pre-ltirr 8239

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-inn 9238  df-n0 9497  df-xnn0 9564

This theorem is referenced by:  xnn0le2is012  10199