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Theorem xnn0lenn0nn0 9588
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
StepHypRef Expression
1 elxnn0 8993 . . 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 3900 . . . . . . 7  |-  ( M  = +oo  ->  ( M  <_  N  <-> +oo  <_  N
) )
43adantr 272 . . . . . 6  |-  ( ( M  = +oo  /\  N  e.  NN0 )  -> 
( M  <_  N  <-> +oo 
<_  N ) )
5 nn0re 8937 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  RR )
65rexrd 7779 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e. 
RR* )
7 xgepnf 9539 . . . . . . . . 9  |-  ( N  e.  RR*  ->  ( +oo  <_  N  <->  N  = +oo ) )
86, 7syl 14 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( +oo  <_  N  <->  N  = +oo ) )
9 pnfnre 7771 . . . . . . . . 9  |- +oo  e/  RR
10 eleq1 2178 . . . . . . . . . . 11  |-  ( N  = +oo  ->  ( N  e.  NN0  <-> +oo  e.  NN0 ) )
11 nn0re 8937 . . . . . . . . . . . 12  |-  ( +oo  e.  NN0  -> +oo  e.  RR )
12 elnelall 2390 . . . . . . . . . . . 12  |-  ( +oo  e.  RR  ->  ( +oo  e/  RR  ->  M  e.  NN0 ) )
1311, 12syl 14 . . . . . . . . . . 11  |-  ( +oo  e.  NN0  ->  ( +oo  e/  RR  ->  M  e.  NN0 ) )
1410, 13syl6bi 162 . . . . . . . . . 10  |-  ( N  = +oo  ->  ( N  e.  NN0  ->  ( +oo  e/  RR  ->  M  e.  NN0 ) ) )
1514com13 80 . . . . . . . . 9  |-  ( +oo  e/  RR  ->  ( N  e.  NN0  ->  ( N  = +oo  ->  M  e.  NN0 ) ) )
169, 15ax-mp 5 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  = +oo  ->  M  e.  NN0 ) )
178, 16sylbid 149 . . . . . . 7  |-  ( N  e.  NN0  ->  ( +oo  <_  N  ->  M  e.  NN0 ) )
1817adantl 273 . . . . . 6  |-  ( ( M  = +oo  /\  N  e.  NN0 )  -> 
( +oo  <_  N  ->  M  e.  NN0 ) )
194, 18sylbid 149 . . . . 5  |-  ( ( M  = +oo  /\  N  e.  NN0 )  -> 
( M  <_  N  ->  M  e.  NN0 )
)
2019ex 114 . . . 4  |-  ( M  = +oo  ->  ( N  e.  NN0  ->  ( M  <_  N  ->  M  e.  NN0 ) ) )
212, 20jaoi 688 . . 3  |-  ( ( M  e.  NN0  \/  M  = +oo )  ->  ( N  e.  NN0  ->  ( M  <_  N  ->  M  e.  NN0 )
) )
221, 21sylbi 120 . 2  |-  ( M  e. NN0*  ->  ( N  e. 
NN0  ->  ( M  <_  N  ->  M  e.  NN0 ) ) )
23223imp 1158 1  |-  ( ( M  e. NN0*  /\  N  e.  NN0  /\  M  <_  N )  ->  M  e.  NN0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463    e/ wnel 2378   class class class wbr 3897   RRcr 7583   +oocpnf 7761   RR*cxr 7763    <_ cle 7765   NN0cn0 8928  NN0*cxnn0 8991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1re 7678  ax-addrcl 7681  ax-rnegex 7693  ax-pre-ltirr 7696
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-inn 8678  df-n0 8929  df-xnn0 8992
This theorem is referenced by:  xnn0le2is012  9589
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