Theorem xnn0lenn0nn0 10267

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 9632	. . 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 4133	. . . . . . 7 |- ( M = +oo -> ( M <_ N <-> +oo <_ N ) )
4	3	adantr 276	. . . . . 6 |- ( ( M = +oo /\ N e. NN0 ) -> ( M <_ N <-> +oo <_ N ) )
5		nn0re 9572	. . . . . . . . . 10 |- ( N e. NN0 -> N e. RR )
6	5	rexrd 8375	. . . . . . . . 9 |- ( N e. NN0 -> N e. RR* )
7		xgepnf 10218	. . . . . . . . 9 |- ( N e. RR* -> ( +oo <_ N <-> N = +oo ) )
8	6, 7	syl 14	. . . . . . . 8 |- ( N e. NN0 -> ( +oo <_ N <-> N = +oo ) )
9		pnfnre 8367	. . . . . . . . 9 |- +oo e/ RR
10		eleq1 2301	. . . . . . . . . . 11 |- ( N = +oo -> ( N e. NN0 <-> +oo e. NN0 ) )
11		nn0re 9572	. . . . . . . . . . . 12 |- ( +oo e. NN0 -> +oo e. RR )
12		elnelall 2527	. . . . . . . . . . . 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 728	. . 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 1224	1 |- ( ( M e. NN0* /\ N e. NN0 /\ M <_ N ) -> M e. NN0 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   e/ wnel 2515   class class class wbr 4130   RRcr 8178   +oocpnf 8357   RR*cxr 8359   <_ cle 8361   NN0cn0 9563  NN0*cxnn0 9630

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276  ax-rnegex 8288  ax-pre-ltirr 8291

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-inn 9305  df-n0 9564  df-xnn0 9631

This theorem is used by:  xnn0le2is012  10268