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Theorem xnn0letri 9822
Description: Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
Assertion
Ref Expression
xnn0letri  |-  ( ( A  e. NN0*  /\  B  e. NN0* )  ->  ( A  <_  B  \/  B  <_  A ) )

Proof of Theorem xnn0letri
StepHypRef Expression
1 simpr 110 . . . . 5  |-  ( ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  e.  NN0 )  /\  A  e.  NN0 )  ->  A  e.  NN0 )
21nn0zd 9392 . . . 4  |-  ( ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  e.  NN0 )  /\  A  e.  NN0 )  ->  A  e.  ZZ )
3 simplr 528 . . . . 5  |-  ( ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  e.  NN0 )  /\  A  e.  NN0 )  ->  B  e.  NN0 )
43nn0zd 9392 . . . 4  |-  ( ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  e.  NN0 )  /\  A  e.  NN0 )  ->  B  e.  ZZ )
5 zletric 9316 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <_  B  \/  B  <_  A ) )
62, 4, 5syl2anc 411 . . 3  |-  ( ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  e.  NN0 )  /\  A  e.  NN0 )  -> 
( A  <_  B  \/  B  <_  A ) )
7 xnn0xr 9263 . . . . . . 7  |-  ( B  e. NN0*  ->  B  e.  RR* )
8 pnfge 9808 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_ +oo )
97, 8syl 14 . . . . . 6  |-  ( B  e. NN0*  ->  B  <_ +oo )
109ad3antlr 493 . . . . 5  |-  ( ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  e.  NN0 )  /\  A  = +oo )  ->  B  <_ +oo )
11 simpr 110 . . . . 5  |-  ( ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  e.  NN0 )  /\  A  = +oo )  ->  A  = +oo )
1210, 11breqtrrd 4046 . . . 4  |-  ( ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  e.  NN0 )  /\  A  = +oo )  ->  B  <_  A )
1312olcd 735 . . 3  |-  ( ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  e.  NN0 )  /\  A  = +oo )  ->  ( A  <_  B  \/  B  <_  A ) )
14 elxnn0 9260 . . . . 5  |-  ( A  e. NN0* 
<->  ( A  e.  NN0  \/  A  = +oo )
)
1514biimpi 120 . . . 4  |-  ( A  e. NN0*  ->  ( A  e. 
NN0  \/  A  = +oo ) )
1615ad2antrr 488 . . 3  |-  ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  e.  NN0 )  ->  ( A  e.  NN0  \/  A  = +oo ) )
176, 13, 16mpjaodan 799 . 2  |-  ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  e.  NN0 )  ->  ( A  <_  B  \/  B  <_  A ) )
18 xnn0xr 9263 . . . . . 6  |-  ( A  e. NN0*  ->  A  e.  RR* )
1918ad2antrr 488 . . . . 5  |-  ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  = +oo )  ->  A  e.  RR* )
20 pnfge 9808 . . . . 5  |-  ( A  e.  RR*  ->  A  <_ +oo )
2119, 20syl 14 . . . 4  |-  ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  = +oo )  ->  A  <_ +oo )
22 simpr 110 . . . 4  |-  ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  = +oo )  ->  B  = +oo )
2321, 22breqtrrd 4046 . . 3  |-  ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  = +oo )  ->  A  <_  B )
2423orcd 734 . 2  |-  ( ( ( A  e. NN0*  /\  B  e. NN0* )  /\  B  = +oo )  ->  ( A  <_  B  \/  B  <_  A ) )
25 elxnn0 9260 . . . 4  |-  ( B  e. NN0* 
<->  ( B  e.  NN0  \/  B  = +oo )
)
2625biimpi 120 . . 3  |-  ( B  e. NN0*  ->  ( B  e. 
NN0  \/  B  = +oo ) )
2726adantl 277 . 2  |-  ( ( A  e. NN0*  /\  B  e. NN0* )  ->  ( B  e.  NN0  \/  B  = +oo ) )
2817, 24, 27mpjaodan 799 1  |-  ( ( A  e. NN0*  /\  B  e. NN0* )  ->  ( A  <_  B  \/  B  <_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 709    = wceq 1364    e. wcel 2160   class class class wbr 4018   +oocpnf 8008   RR*cxr 8010    <_ cle 8012   NN0cn0 9195  NN0*cxnn0 9258   ZZcz 9272
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-ltadd 7946
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-inn 8939  df-n0 9196  df-xnn0 9259  df-z 9273
This theorem is referenced by:  pcgcd  12347
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