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Theorem xnn0letri 9878
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 9446 . . . 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 9446 . . . 4 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> B e. ZZ )
5 zletric 9370 . . . 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 9317 . . . . . . 7 |- ( B e. NN0* -> B e. RR* )
8 pnfge 9864 . . . . . . 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 4061 . . . 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 9314 . . . . 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 9317 . . . . . 6 |- ( A e. NN0* -> A e. RR* )
1918ad2antrr 488 . . . . 5 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> A e. RR* )
20 pnfge 9864 . . . . 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 4061 . . 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 9314 . . . 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 2167   class class class wbr 4033   +oocpnf 8058   RR*cxr 8060   <_ cle 8062   NN0cn0 9249  NN0*cxnn0 9312   ZZcz 9326
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-xnn0 9313  df-z 9327
This theorem is referenced by:  pcgcd  12498
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