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Theorem xnn0letri 9760
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 109 . . . . 5 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> A e. NN0 )
21nn0zd 9332 . . . 4 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> A e. ZZ )
3 simplr 525 . . . . 5 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> B e. NN0 )
43nn0zd 9332 . . . 4 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> B e. ZZ )
5 zletric 9256 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B <_ A ) )
62, 4, 5syl2anc 409 . . 3 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> ( A <_ B \/ B <_ A ) )
7 xnn0xr 9203 . . . . . . 7 |- ( B e. NN0* -> B e. RR* )
8 pnfge 9746 . . . . . . 7 |- ( B e. RR* -> B <_ +oo )
97, 8syl 14 . . . . . 6 |- ( B e. NN0* -> B <_ +oo )
109ad3antlr 490 . . . . 5 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> B <_ +oo )
11 simpr 109 . . . . 5 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> A = +oo )
1210, 11breqtrrd 4017 . . . 4 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> B <_ A )
1312olcd 729 . . 3 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> ( A <_ B \/ B <_ A ) )
14 elxnn0 9200 . . . . 5 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
1514biimpi 119 . . . 4 |- ( A e. NN0* -> ( A e. NN0 \/ A = +oo ) )
1615ad2antrr 485 . . 3 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) -> ( A e. NN0 \/ A = +oo ) )
176, 13, 16mpjaodan 793 . 2 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) -> ( A <_ B \/ B <_ A ) )
18 xnn0xr 9203 . . . . . 6 |- ( A e. NN0* -> A e. RR* )
1918ad2antrr 485 . . . . 5 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> A e. RR* )
20 pnfge 9746 . . . . 5 |- ( A e. RR* -> A <_ +oo )
2119, 20syl 14 . . . 4 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> A <_ +oo )
22 simpr 109 . . . 4 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> B = +oo )
2321, 22breqtrrd 4017 . . 3 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> A <_ B )
2423orcd 728 . 2 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> ( A <_ B \/ B <_ A ) )
25 elxnn0 9200 . . . 4 |- ( B e. NN0* <-> ( B e. NN0 \/ B = +oo ) )
2625biimpi 119 . . 3 |- ( B e. NN0* -> ( B e. NN0 \/ B = +oo ) )
2726adantl 275 . 2 |- ( ( A e. NN0* /\ B e. NN0* ) -> ( B e. NN0 \/ B = +oo ) )
2817, 24, 27mpjaodan 793 1 |- ( ( A e. NN0* /\ B e. NN0* ) -> ( A <_ B \/ B <_ A ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   \/ wo 703   = wceq 1348   e. wcel 2141   class class class wbr 3989   +oocpnf 7951   RR*cxr 7953   <_ cle 7955   NN0cn0 9135  NN0*cxnn0 9198   ZZcz 9212
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-xnn0 9199  df-z 9213
This theorem is referenced by:  pcgcd  12282
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