Theorem xnn0lenn0nn0 10090

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 9457	. . 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 4089	. . . . . . 7 |- ( M = +oo -> ( M <_ N <-> +oo <_ N ) )
4	3	adantr 276	. . . . . 6 |- ( ( M = +oo /\ N e. NN0 ) -> ( M <_ N <-> +oo <_ N ) )
5		nn0re 9401	. . . . . . . . . 10 |- ( N e. NN0 -> N e. RR )
6	5	rexrd 8219	. . . . . . . . 9 |- ( N e. NN0 -> N e. RR* )
7		xgepnf 10041	. . . . . . . . 9 |- ( N e. RR* -> ( +oo <_ N <-> N = +oo ) )
8	6, 7	syl 14	. . . . . . . 8 |- ( N e. NN0 -> ( +oo <_ N <-> N = +oo ) )
9		pnfnre 8211	. . . . . . . . 9 |- +oo e/ RR
10		eleq1 2292	. . . . . . . . . . 11 |- ( N = +oo -> ( N e. NN0 <-> +oo e. NN0 ) )
11		nn0re 9401	. . . . . . . . . . . 12 |- ( +oo e. NN0 -> +oo e. RR )
12		elnelall 2507	. . . . . . . . . . . 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 721	. . 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 1217	1 |- ( ( M e. NN0* /\ N e. NN0 /\ M <_ N ) -> M e. NN0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   e/ wnel 2495   class class class wbr 4086   RRcr 8021   +oocpnf 8201   RR*cxr 8203   <_ cle 8205   NN0cn0 9392  NN0*cxnn0 9455

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119  ax-rnegex 8131  ax-pre-ltirr 8134

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  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 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-xp 4729  df-cnv 4731  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-inn 9134  df-n0 9393  df-xnn0 9456

This theorem is referenced by:  xnn0le2is012  10091