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Mirrors > Home > ILE Home > Th. List > nnawordex | Unicode version |
Description: Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnawordex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nntri3or 6357 | . . . . 5 | |
2 | 1 | 3adant3 986 | . . . 4 |
3 | nnaordex 6391 | . . . . . . 7 | |
4 | simpr 109 | . . . . . . . 8 | |
5 | 4 | reximi 2506 | . . . . . . 7 |
6 | 3, 5 | syl6bi 162 | . . . . . 6 |
7 | 6 | 3adant3 986 | . . . . 5 |
8 | nna0 6338 | . . . . . . . 8 | |
9 | 8 | 3ad2ant1 987 | . . . . . . 7 |
10 | eqeq2 2127 | . . . . . . 7 | |
11 | 9, 10 | syl5ibcom 154 | . . . . . 6 |
12 | peano1 4478 | . . . . . . 7 | |
13 | oveq2 5750 | . . . . . . . . 9 | |
14 | 13 | eqeq1d 2126 | . . . . . . . 8 |
15 | 14 | rspcev 2763 | . . . . . . 7 |
16 | 12, 15 | mpan 420 | . . . . . 6 |
17 | 11, 16 | syl6 33 | . . . . 5 |
18 | nntri1 6360 | . . . . . . 7 | |
19 | 18 | biimp3a 1308 | . . . . . 6 |
20 | 19 | pm2.21d 593 | . . . . 5 |
21 | 7, 17, 20 | 3jaod 1267 | . . . 4 |
22 | 2, 21 | mpd 13 | . . 3 |
23 | 22 | 3expia 1168 | . 2 |
24 | nnaword1 6377 | . . . . 5 | |
25 | sseq2 3091 | . . . . 5 | |
26 | 24, 25 | syl5ibcom 154 | . . . 4 |
27 | 26 | rexlimdva 2526 | . . 3 |
28 | 27 | adantr 274 | . 2 |
29 | 23, 28 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 946 w3a 947 wceq 1316 wcel 1465 wrex 2394 wss 3041 c0 3333 com 4474 (class class class)co 5742 coa 6278 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-oadd 6285 |
This theorem is referenced by: prarloclemn 7275 |
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