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Mirrors > Home > ILE Home > Th. List > ennnfonelemg | Unicode version |
Description: Lemma for ennnfone 12295. Closure for . (Contributed by Jim Kingdon, 20-Jul-2023.) |
Ref | Expression |
---|---|
ennnfonelemh.dceq | DECID |
ennnfonelemh.f | |
ennnfonelemh.ne | |
ennnfonelemh.g | |
ennnfonelemh.n | frec |
ennnfonelemh.j | |
ennnfonelemh.h |
Ref | Expression |
---|---|
ennnfonelemg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ennnfonelemh.g | . . . 4 | |
2 | 1 | a1i 9 | . . 3 |
3 | simpr 109 | . . . . . . 7 | |
4 | 3 | fveq2d 5484 | . . . . . 6 |
5 | 3 | imaeq2d 4940 | . . . . . 6 |
6 | 4, 5 | eleq12d 2235 | . . . . 5 |
7 | simpl 108 | . . . . 5 | |
8 | 7 | dmeqd 4800 | . . . . . . . 8 |
9 | 8, 4 | opeq12d 3760 | . . . . . . 7 |
10 | 9 | sneqd 3583 | . . . . . 6 |
11 | 7, 10 | uneq12d 3272 | . . . . 5 |
12 | 6, 7, 11 | ifbieq12d 3541 | . . . 4 |
13 | 12 | adantl 275 | . . 3 |
14 | ssrab2 3222 | . . . 4 | |
15 | simprl 521 | . . . 4 | |
16 | 14, 15 | sseldi 3135 | . . 3 |
17 | simprr 522 | . . 3 | |
18 | simplrl 525 | . . . 4 | |
19 | dmeq 4798 | . . . . . 6 | |
20 | 19 | eleq1d 2233 | . . . . 5 |
21 | omex 4564 | . . . . . . . 8 | |
22 | ennnfonelemh.f | . . . . . . . 8 | |
23 | focdmex 10689 | . . . . . . . 8 | |
24 | 21, 22, 23 | sylancr 411 | . . . . . . 7 |
25 | 24 | ad2antrr 480 | . . . . . 6 |
26 | 21 | a1i 9 | . . . . . 6 |
27 | simplrl 525 | . . . . . . . 8 | |
28 | elrabi 2874 | . . . . . . . . . 10 | |
29 | elpmi 6624 | . . . . . . . . . 10 | |
30 | 28, 29 | syl 14 | . . . . . . . . 9 |
31 | 30 | simpld 111 | . . . . . . . 8 |
32 | 27, 31 | syl 14 | . . . . . . 7 |
33 | dmeq 4798 | . . . . . . . . . . 11 | |
34 | 33 | eleq1d 2233 | . . . . . . . . . 10 |
35 | 34 | elrab 2877 | . . . . . . . . 9 |
36 | 35 | simprbi 273 | . . . . . . . 8 |
37 | 27, 36 | syl 14 | . . . . . . 7 |
38 | nnord 4583 | . . . . . . . . 9 | |
39 | 37, 38 | syl 14 | . . . . . . . 8 |
40 | ordirr 4513 | . . . . . . . 8 | |
41 | 39, 40 | syl 14 | . . . . . . 7 |
42 | 22 | adantr 274 | . . . . . . . . . 10 |
43 | fof 5404 | . . . . . . . . . 10 | |
44 | 42, 43 | syl 14 | . . . . . . . . 9 |
45 | 44, 17 | ffvelrnd 5615 | . . . . . . . 8 |
46 | 45 | adantr 274 | . . . . . . 7 |
47 | fsnunf 5679 | . . . . . . 7 | |
48 | 32, 37, 41, 46, 47 | syl121anc 1232 | . . . . . 6 |
49 | df-suc 4343 | . . . . . . . . 9 | |
50 | peano2 4566 | . . . . . . . . 9 | |
51 | 49, 50 | eqeltrrid 2252 | . . . . . . . 8 |
52 | 37, 51 | syl 14 | . . . . . . 7 |
53 | omelon 4580 | . . . . . . . 8 | |
54 | 53 | onelssi 4401 | . . . . . . 7 |
55 | 52, 54 | syl 14 | . . . . . 6 |
56 | elpm2r 6623 | . . . . . 6 | |
57 | 25, 26, 48, 55, 56 | syl22anc 1228 | . . . . 5 |
58 | 48 | fdmd 5338 | . . . . . 6 |
59 | 58, 52 | eqeltrd 2241 | . . . . 5 |
60 | 20, 57, 59 | elrabd 2879 | . . . 4 |
61 | ennnfonelemh.dceq | . . . . . 6 DECID | |
62 | 61 | adantr 274 | . . . . 5 DECID |
63 | 62, 42, 17 | ennnfonelemdc 12269 | . . . 4 DECID |
64 | 18, 60, 63 | ifcldadc 3544 | . . 3 |
65 | 2, 13, 16, 17, 64 | ovmpod 5960 | . 2 |
66 | 65, 64 | eqeltrd 2241 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 DECID wdc 824 wceq 1342 wcel 2135 wne 2334 wral 2442 wrex 2443 crab 2446 cvv 2721 cun 3109 wss 3111 c0 3404 cif 3515 csn 3570 cop 3573 cmpt 4037 word 4334 csuc 4337 com 4561 ccnv 4597 cdm 4598 cima 4601 wf 5178 wfo 5180 cfv 5182 (class class class)co 5836 cmpo 5838 freccfrec 6349 cpm 6606 cc0 7744 c1 7745 caddc 7747 cmin 8060 cn0 9105 cz 9182 cseq 10370 |
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 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 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-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pm 6608 |
This theorem is referenced by: ennnfonelemh 12274 ennnfonelem0 12275 ennnfonelemp1 12276 ennnfonelemom 12278 |
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