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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 6441 | . . . . 5 | |
2 | 1 | 3adant3 1002 | . . . 4 |
3 | nnaordex 6475 | . . . . . . 7 | |
4 | simpr 109 | . . . . . . . 8 | |
5 | 4 | reximi 2554 | . . . . . . 7 |
6 | 3, 5 | syl6bi 162 | . . . . . 6 |
7 | 6 | 3adant3 1002 | . . . . 5 |
8 | nna0 6422 | . . . . . . . 8 | |
9 | 8 | 3ad2ant1 1003 | . . . . . . 7 |
10 | eqeq2 2167 | . . . . . . 7 | |
11 | 9, 10 | syl5ibcom 154 | . . . . . 6 |
12 | peano1 4554 | . . . . . . 7 | |
13 | oveq2 5833 | . . . . . . . . 9 | |
14 | 13 | eqeq1d 2166 | . . . . . . . 8 |
15 | 14 | rspcev 2816 | . . . . . . 7 |
16 | 12, 15 | mpan 421 | . . . . . 6 |
17 | 11, 16 | syl6 33 | . . . . 5 |
18 | nntri1 6444 | . . . . . . 7 | |
19 | 18 | biimp3a 1327 | . . . . . 6 |
20 | 19 | pm2.21d 609 | . . . . 5 |
21 | 7, 17, 20 | 3jaod 1286 | . . . 4 |
22 | 2, 21 | mpd 13 | . . 3 |
23 | 22 | 3expia 1187 | . 2 |
24 | nnaword1 6461 | . . . . 5 | |
25 | sseq2 3152 | . . . . 5 | |
26 | 24, 25 | syl5ibcom 154 | . . . 4 |
27 | 26 | rexlimdva 2574 | . . 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 962 w3a 963 wceq 1335 wcel 2128 wrex 2436 wss 3102 c0 3394 com 4550 (class class class)co 5825 coa 6361 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-1o 6364 df-oadd 6368 |
This theorem is referenced by: prarloclemn 7420 |
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