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Mirrors > Home > ILE Home > Th. List > nnaord | Unicode version |
Description: Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers, and its converse. (Contributed by NM, 7-Mar-1996.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnaord |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnaordi 6281 |
. . 3
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2 | 1 | 3adant1 962 |
. 2
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3 | oveq2 5674 |
. . . . . 6
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4 | 3 | a1i 9 |
. . . . 5
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5 | nnaordi 6281 |
. . . . . 6
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6 | 5 | 3adant2 963 |
. . . . 5
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7 | 4, 6 | orim12d 736 |
. . . 4
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8 | 7 | con3d 597 |
. . 3
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9 | df-3an 927 |
. . . . . 6
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10 | ancom 263 |
. . . . . 6
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11 | anandi 558 |
. . . . . 6
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12 | 9, 10, 11 | 3bitri 205 |
. . . . 5
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13 | nnacl 6255 |
. . . . . 6
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14 | nnacl 6255 |
. . . . . 6
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15 | 13, 14 | anim12i 332 |
. . . . 5
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16 | 12, 15 | sylbi 120 |
. . . 4
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17 | nntri2 6269 |
. . . 4
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18 | 16, 17 | syl 14 |
. . 3
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19 | nntri2 6269 |
. . . 4
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20 | 19 | 3adant3 964 |
. . 3
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21 | 8, 18, 20 | 3imtr4d 202 |
. 2
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22 | 2, 21 | impbid 128 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-oadd 6199 |
This theorem is referenced by: nnaordr 6283 nnaordex 6300 ltapig 6958 1lt2pi 6960 |
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