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Mirrors > Home > ILE Home > Th. List > xnn0letri | Unicode version |
Description: Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.) |
Ref | Expression |
---|---|
xnn0letri | NN0* NN0* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 NN0* NN0* | |
2 | 1 | nn0zd 9321 | . . . 4 NN0* NN0* |
3 | simplr 525 | . . . . 5 NN0* NN0* | |
4 | 3 | nn0zd 9321 | . . . 4 NN0* NN0* |
5 | zletric 9245 | . . . 4 | |
6 | 2, 4, 5 | syl2anc 409 | . . 3 NN0* NN0* |
7 | xnn0xr 9192 | . . . . . . 7 NN0* | |
8 | pnfge 9735 | . . . . . . 7 | |
9 | 7, 8 | syl 14 | . . . . . 6 NN0* |
10 | 9 | ad3antlr 490 | . . . . 5 NN0* NN0* |
11 | simpr 109 | . . . . 5 NN0* NN0* | |
12 | 10, 11 | breqtrrd 4015 | . . . 4 NN0* NN0* |
13 | 12 | olcd 729 | . . 3 NN0* NN0* |
14 | elxnn0 9189 | . . . . 5 NN0* | |
15 | 14 | biimpi 119 | . . . 4 NN0* |
16 | 15 | ad2antrr 485 | . . 3 NN0* NN0* |
17 | 6, 13, 16 | mpjaodan 793 | . 2 NN0* NN0* |
18 | xnn0xr 9192 | . . . . . 6 NN0* | |
19 | 18 | ad2antrr 485 | . . . . 5 NN0* NN0* |
20 | pnfge 9735 | . . . . 5 | |
21 | 19, 20 | syl 14 | . . . 4 NN0* NN0* |
22 | simpr 109 | . . . 4 NN0* NN0* | |
23 | 21, 22 | breqtrrd 4015 | . . 3 NN0* NN0* |
24 | 23 | orcd 728 | . 2 NN0* NN0* |
25 | elxnn0 9189 | . . . 4 NN0* | |
26 | 25 | biimpi 119 | . . 3 NN0* |
27 | 26 | adantl 275 | . 2 NN0* NN0* |
28 | 17, 24, 27 | mpjaodan 793 | 1 NN0* NN0* |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 703 wceq 1348 wcel 2141 class class class wbr 3987 cpnf 7940 cxr 7942 cle 7944 cn0 9124 NN0*cxnn0 9187 cz 9201 |
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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-addcom 7863 ax-addass 7865 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-0id 7871 ax-rnegex 7872 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-ltadd 7879 |
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 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-inn 8868 df-n0 9125 df-xnn0 9188 df-z 9202 |
This theorem is referenced by: pcgcd 12271 |
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