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Mirrors > Home > ILE Home > Th. List > frec2uzltd | Unicode version |
Description: Less-than relation for (see frec2uz0d 10325). (Contributed by Jim Kingdon, 16-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | |
frec2uz.2 | frec |
frec2uzzd.a | |
frec2uzltd.b |
Ref | Expression |
---|---|
frec2uzltd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uzltd.b | . 2 | |
2 | eleq2 2228 | . . . . 5 | |
3 | fveq2 5481 | . . . . . 6 | |
4 | 3 | breq2d 3989 | . . . . 5 |
5 | 2, 4 | imbi12d 233 | . . . 4 |
6 | 5 | imbi2d 229 | . . 3 |
7 | eleq2 2228 | . . . . 5 | |
8 | fveq2 5481 | . . . . . 6 | |
9 | 8 | breq2d 3989 | . . . . 5 |
10 | 7, 9 | imbi12d 233 | . . . 4 |
11 | 10 | imbi2d 229 | . . 3 |
12 | eleq2 2228 | . . . . 5 | |
13 | fveq2 5481 | . . . . . 6 | |
14 | 13 | breq2d 3989 | . . . . 5 |
15 | 12, 14 | imbi12d 233 | . . . 4 |
16 | 15 | imbi2d 229 | . . 3 |
17 | eleq2 2228 | . . . . 5 | |
18 | fveq2 5481 | . . . . . 6 | |
19 | 18 | breq2d 3989 | . . . . 5 |
20 | 17, 19 | imbi12d 233 | . . . 4 |
21 | 20 | imbi2d 229 | . . 3 |
22 | noel 3409 | . . . . 5 | |
23 | 22 | pm2.21i 636 | . . . 4 |
24 | 23 | a1i 9 | . . 3 |
25 | id 19 | . . . . . . 7 | |
26 | fveq2 5481 | . . . . . . . 8 | |
27 | 26 | a1i 9 | . . . . . . 7 |
28 | 25, 27 | orim12d 776 | . . . . . 6 |
29 | elsuc2g 4378 | . . . . . . . . 9 | |
30 | 29 | bicomd 140 | . . . . . . . 8 |
31 | 30 | adantr 274 | . . . . . . 7 |
32 | frec2uz.1 | . . . . . . . . . . 11 | |
33 | 32 | adantl 275 | . . . . . . . . . 10 |
34 | frec2uz.2 | . . . . . . . . . 10 frec | |
35 | simpl 108 | . . . . . . . . . 10 | |
36 | 33, 34, 35 | frec2uzsucd 10327 | . . . . . . . . 9 |
37 | 36 | breq2d 3989 | . . . . . . . 8 |
38 | frec2uzzd.a | . . . . . . . . . . 11 | |
39 | 38 | adantl 275 | . . . . . . . . . 10 |
40 | 33, 34, 39 | frec2uzuzd 10328 | . . . . . . . . 9 |
41 | 33, 34, 35 | frec2uzuzd 10328 | . . . . . . . . 9 |
42 | eluzelz 9467 | . . . . . . . . . 10 | |
43 | eluzelz 9467 | . . . . . . . . . 10 | |
44 | zleltp1 9238 | . . . . . . . . . 10 | |
45 | 42, 43, 44 | syl2an 287 | . . . . . . . . 9 |
46 | 40, 41, 45 | syl2anc 409 | . . . . . . . 8 |
47 | 33, 34, 39 | frec2uzzd 10326 | . . . . . . . . 9 |
48 | 33, 34, 35 | frec2uzzd 10326 | . . . . . . . . 9 |
49 | zleloe 9230 | . . . . . . . . 9 | |
50 | 47, 48, 49 | syl2anc 409 | . . . . . . . 8 |
51 | 37, 46, 50 | 3bitr2rd 216 | . . . . . . 7 |
52 | 31, 51 | imbi12d 233 | . . . . . 6 |
53 | 28, 52 | syl5ib 153 | . . . . 5 |
54 | 53 | ex 114 | . . . 4 |
55 | 54 | a2d 26 | . . 3 |
56 | 6, 11, 16, 21, 24, 55 | finds 4572 | . 2 |
57 | 1, 56 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 c0 3405 class class class wbr 3977 cmpt 4038 csuc 4338 com 4562 cfv 5183 (class class class)co 5837 freccfrec 6350 c1 7746 caddc 7748 clt 7925 cle 7926 cz 9183 cuz 9458 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-addass 7847 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-0id 7853 ax-rnegex 7854 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-ltadd 7861 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-iord 4339 df-on 4341 df-ilim 4342 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-recs 6265 df-frec 6351 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-inn 8850 df-n0 9107 df-z 9184 df-uz 9459 |
This theorem is referenced by: frec2uzlt2d 10330 frec2uzf1od 10332 ennnfonelemex 12310 ennnfonelemnn0 12318 |
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