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Mirrors > Home > ILE Home > Th. List > ltexpi | Unicode version |
Description: Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) |
Ref | Expression |
---|---|
ltexpi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 6293 |
. . 3
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2 | pinn 6293 |
. . 3
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3 | nnaordex 6036 |
. . 3
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4 | 1, 2, 3 | syl2an 273 |
. 2
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5 | ltpiord 6303 |
. 2
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6 | addpiord 6300 |
. . . . . . 7
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7 | 6 | eqeq1d 2045 |
. . . . . 6
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8 | 7 | pm5.32da 425 |
. . . . 5
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9 | elni2 6298 |
. . . . . . 7
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10 | 9 | anbi1i 431 |
. . . . . 6
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11 | anass 381 |
. . . . . 6
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12 | 10, 11 | bitri 173 |
. . . . 5
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13 | 8, 12 | syl6bb 185 |
. . . 4
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14 | 13 | rexbidv2 2323 |
. . 3
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15 | 14 | adantr 261 |
. 2
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16 | 4, 5, 15 | 3bitr4d 209 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-oadd 5944 df-ni 6288 df-pli 6289 df-lti 6291 |
This theorem is referenced by: ltexnqq 6391 |
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