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Mirrors > Home > ILE Home > Th. List > fidcenumlemrks | Unicode version |
Description: Lemma for fidcenum 6812. Induction step for fidcenumlemrk 6810. (Contributed by Jim Kingdon, 20-Oct-2022.) |
Ref | Expression |
---|---|
fidcenumlemr.dc | DECID |
fidcenumlemr.f | |
fidcenumlemrks.j | |
fidcenumlemrks.jn | |
fidcenumlemrks.h | |
fidcenumlemrks.x |
Ref | Expression |
---|---|
fidcenumlemrks |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 | |
2 | elun1 3213 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | df-suc 4263 | . . . . . . 7 | |
5 | 4 | imaeq2i 4849 | . . . . . 6 |
6 | imaundi 4921 | . . . . . 6 | |
7 | 5, 6 | eqtri 2138 | . . . . 5 |
8 | 7 | eleq2i 2184 | . . . 4 |
9 | 3, 8 | sylibr 133 | . . 3 |
10 | 9 | orcd 707 | . 2 |
11 | simpr 109 | . . . . . . 7 | |
12 | fidcenumlemrks.x | . . . . . . . . . 10 | |
13 | elsng 3512 | . . . . . . . . . 10 | |
14 | 12, 13 | syl 14 | . . . . . . . . 9 |
15 | fidcenumlemr.f | . . . . . . . . . . . 12 | |
16 | fofn 5317 | . . . . . . . . . . . 12 | |
17 | 15, 16 | syl 14 | . . . . . . . . . . 11 |
18 | fidcenumlemrks.jn | . . . . . . . . . . . 12 | |
19 | fidcenumlemrks.j | . . . . . . . . . . . . 13 | |
20 | sucidg 4308 | . . . . . . . . . . . . 13 | |
21 | 19, 20 | syl 14 | . . . . . . . . . . . 12 |
22 | 18, 21 | sseldd 3068 | . . . . . . . . . . 11 |
23 | fnsnfv 5448 | . . . . . . . . . . 11 | |
24 | 17, 22, 23 | syl2anc 408 | . . . . . . . . . 10 |
25 | 24 | eleq2d 2187 | . . . . . . . . 9 |
26 | 14, 25 | bitr3d 189 | . . . . . . . 8 |
27 | 26 | ad2antrr 479 | . . . . . . 7 |
28 | 11, 27 | mpbid 146 | . . . . . 6 |
29 | elun2 3214 | . . . . . 6 | |
30 | 28, 29 | syl 14 | . . . . 5 |
31 | 30, 8 | sylibr 133 | . . . 4 |
32 | 31 | orcd 707 | . . 3 |
33 | simplr 504 | . . . . . . 7 | |
34 | simpr 109 | . . . . . . . 8 | |
35 | 26 | ad2antrr 479 | . . . . . . . 8 |
36 | 34, 35 | mtbid 646 | . . . . . . 7 |
37 | ioran 726 | . . . . . . 7 | |
38 | 33, 36, 37 | sylanbrc 413 | . . . . . 6 |
39 | elun 3187 | . . . . . 6 | |
40 | 38, 39 | sylnibr 651 | . . . . 5 |
41 | 40, 8 | sylnibr 651 | . . . 4 |
42 | 41 | olcd 708 | . . 3 |
43 | fof 5315 | . . . . . . . 8 | |
44 | 15, 43 | syl 14 | . . . . . . 7 |
45 | 44, 22 | ffvelrnd 5524 | . . . . . 6 |
46 | fidcenumlemr.dc | . . . . . 6 DECID | |
47 | eqeq1 2124 | . . . . . . . 8 | |
48 | 47 | dcbid 808 | . . . . . . 7 DECID DECID |
49 | eqeq2 2127 | . . . . . . . 8 | |
50 | 49 | dcbid 808 | . . . . . . 7 DECID DECID |
51 | 48, 50 | rspc2va 2777 | . . . . . 6 DECID DECID |
52 | 12, 45, 46, 51 | syl21anc 1200 | . . . . 5 DECID |
53 | exmiddc 806 | . . . . 5 DECID | |
54 | 52, 53 | syl 14 | . . . 4 |
55 | 54 | adantr 274 | . . 3 |
56 | 32, 42, 55 | mpjaodan 772 | . 2 |
57 | fidcenumlemrks.h | . 2 | |
58 | 10, 56, 57 | mpjaodan 772 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 DECID wdc 804 wceq 1316 wcel 1465 wral 2393 cun 3039 wss 3041 csn 3497 csuc 4257 com 4474 cima 4512 wfn 5088 wf 5089 wfo 5091 cfv 5093 |
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-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3an 949 df-tru 1319 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-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-suc 4263 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-fo 5099 df-fv 5101 |
This theorem is referenced by: fidcenumlemrk 6810 |
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