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Mirrors > Home > ILE Home > Th. List > ennnfonelemg | Unicode version |
Description: Lemma for ennnfone 11938. 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 5425 | . . . . . 6 |
5 | 3 | imaeq2d 4881 | . . . . . 6 |
6 | 4, 5 | eleq12d 2210 | . . . . 5 |
7 | simpl 108 | . . . . 5 | |
8 | 7 | dmeqd 4741 | . . . . . . . 8 |
9 | 8, 4 | opeq12d 3713 | . . . . . . 7 |
10 | 9 | sneqd 3540 | . . . . . 6 |
11 | 7, 10 | uneq12d 3231 | . . . . 5 |
12 | 6, 7, 11 | ifbieq12d 3498 | . . . 4 |
13 | 12 | adantl 275 | . . 3 |
14 | ssrab2 3182 | . . . 4 | |
15 | simprl 520 | . . . 4 | |
16 | 14, 15 | sseldi 3095 | . . 3 |
17 | simprr 521 | . . 3 | |
18 | simplrl 524 | . . . 4 | |
19 | dmeq 4739 | . . . . . 6 | |
20 | 19 | eleq1d 2208 | . . . . 5 |
21 | omex 4507 | . . . . . . . 8 | |
22 | ennnfonelemh.f | . . . . . . . 8 | |
23 | focdmex 10533 | . . . . . . . 8 | |
24 | 21, 22, 23 | sylancr 410 | . . . . . . 7 |
25 | 24 | ad2antrr 479 | . . . . . 6 |
26 | 21 | a1i 9 | . . . . . 6 |
27 | simplrl 524 | . . . . . . . 8 | |
28 | elrabi 2837 | . . . . . . . . . 10 | |
29 | elpmi 6561 | . . . . . . . . . 10 | |
30 | 28, 29 | syl 14 | . . . . . . . . 9 |
31 | 30 | simpld 111 | . . . . . . . 8 |
32 | 27, 31 | syl 14 | . . . . . . 7 |
33 | dmeq 4739 | . . . . . . . . . . 11 | |
34 | 33 | eleq1d 2208 | . . . . . . . . . 10 |
35 | 34 | elrab 2840 | . . . . . . . . 9 |
36 | 35 | simprbi 273 | . . . . . . . 8 |
37 | 27, 36 | syl 14 | . . . . . . 7 |
38 | nnord 4525 | . . . . . . . . 9 | |
39 | 37, 38 | syl 14 | . . . . . . . 8 |
40 | ordirr 4457 | . . . . . . . 8 | |
41 | 39, 40 | syl 14 | . . . . . . 7 |
42 | 22 | adantr 274 | . . . . . . . . . 10 |
43 | fof 5345 | . . . . . . . . . 10 | |
44 | 42, 43 | syl 14 | . . . . . . . . 9 |
45 | 44, 17 | ffvelrnd 5556 | . . . . . . . 8 |
46 | 45 | adantr 274 | . . . . . . 7 |
47 | fsnunf 5620 | . . . . . . 7 | |
48 | 32, 37, 41, 46, 47 | syl121anc 1221 | . . . . . 6 |
49 | df-suc 4293 | . . . . . . . . 9 | |
50 | peano2 4509 | . . . . . . . . 9 | |
51 | 49, 50 | eqeltrrid 2227 | . . . . . . . 8 |
52 | 37, 51 | syl 14 | . . . . . . 7 |
53 | omelon 4522 | . . . . . . . 8 | |
54 | 53 | onelssi 4351 | . . . . . . 7 |
55 | 52, 54 | syl 14 | . . . . . 6 |
56 | elpm2r 6560 | . . . . . 6 | |
57 | 25, 26, 48, 55, 56 | syl22anc 1217 | . . . . 5 |
58 | 48 | fdmd 5279 | . . . . . 6 |
59 | 58, 52 | eqeltrd 2216 | . . . . 5 |
60 | 20, 57, 59 | elrabd 2842 | . . . 4 |
61 | ennnfonelemh.dceq | . . . . . 6 DECID | |
62 | 61 | adantr 274 | . . . . 5 DECID |
63 | 62, 42, 17 | ennnfonelemdc 11912 | . . . 4 DECID |
64 | 18, 60, 63 | ifcldadc 3501 | . . 3 |
65 | 2, 13, 16, 17, 64 | ovmpod 5898 | . 2 |
66 | 65, 64 | eqeltrd 2216 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 DECID wdc 819 wceq 1331 wcel 1480 wne 2308 wral 2416 wrex 2417 crab 2420 cvv 2686 cun 3069 wss 3071 c0 3363 cif 3474 csn 3527 cop 3530 cmpt 3989 word 4284 csuc 4287 com 4504 ccnv 4538 cdm 4539 cima 4542 wf 5119 wfo 5121 cfv 5123 (class class class)co 5774 cmpo 5776 freccfrec 6287 cpm 6543 cc0 7620 c1 7621 caddc 7623 cmin 7933 cn0 8977 cz 9054 cseq 10218 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pm 6545 |
This theorem is referenced by: ennnfonelemh 11917 ennnfonelem0 11918 ennnfonelemp1 11919 ennnfonelemom 11921 |
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