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Theorem xnn0letri 9872
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 9440 . . . 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 9440 . . . 4 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> B e. ZZ )
5 zletric 9364 . . . 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 9311 . . . . . . 7 |- ( B e. NN0* -> B e. RR* )
8 pnfge 9858 . . . . . . 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 4058 . . . 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 9308 . . . . 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 9311 . . . . . 6 |- ( A e. NN0* -> A e. RR* )
1918ad2antrr 488 . . . . 5 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> A e. RR* )
20 pnfge 9858 . . . . 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 4058 . . 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 9308 . . . 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 2164   class class class wbr 4030   +oocpnf 8053   RR*cxr 8055   <_ cle 8057   NN0cn0 9243  NN0*cxnn0 9306   ZZcz 9320
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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990
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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-xnn0 9307  df-z 9321
This theorem is referenced by:  pcgcd  12470
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