Theorem xnn0lenn0nn0 10022

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 9395	. . 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 4062	. . . . . . 7 |- ( M = +oo -> ( M <_ N <-> +oo <_ N ) )
4	3	adantr 276	. . . . . 6 |- ( ( M = +oo /\ N e. NN0 ) -> ( M <_ N <-> +oo <_ N ) )
5		nn0re 9339	. . . . . . . . . 10 |- ( N e. NN0 -> N e. RR )
6	5	rexrd 8157	. . . . . . . . 9 |- ( N e. NN0 -> N e. RR* )
7		xgepnf 9973	. . . . . . . . 9 |- ( N e. RR* -> ( +oo <_ N <-> N = +oo ) )
8	6, 7	syl 14	. . . . . . . 8 |- ( N e. NN0 -> ( +oo <_ N <-> N = +oo ) )
9		pnfnre 8149	. . . . . . . . 9 |- +oo e/ RR
10		eleq1 2270	. . . . . . . . . . 11 |- ( N = +oo -> ( N e. NN0 <-> +oo e. NN0 ) )
11		nn0re 9339	. . . . . . . . . . . 12 |- ( +oo e. NN0 -> +oo e. RR )
12		elnelall 2485	. . . . . . . . . . . 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 718	. . 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 1196	1 |- ( ( M e. NN0* /\ N e. NN0 /\ M <_ N ) -> M e. NN0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2178   e/ wnel 2473   class class class wbr 4059   RRcr 7959   +oocpnf 8139   RR*cxr 8141   <_ cle 8143   NN0cn0 9330  NN0*cxnn0 9393

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057  ax-rnegex 8069  ax-pre-ltirr 8072

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-inn 9072  df-n0 9331  df-xnn0 9394

This theorem is referenced by:  xnn0le2is012  10023