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Mirrors > Home > ILE Home > Th. List > fidcenumlemrk | Unicode version |
Description: Lemma for fidcenum 6837. (Contributed by Jim Kingdon, 20-Oct-2022.) |
Ref | Expression |
---|---|
fidcenumlemr.dc | DECID |
fidcenumlemr.f | |
fidcenumlemrk.k | |
fidcenumlemrk.kn | |
fidcenumlemrk.x |
Ref | Expression |
---|---|
fidcenumlemrk |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fidcenumlemrk.k | . 2 | |
2 | fidcenumlemrk.kn | . . 3 | |
3 | 2 | ancli 321 | . 2 |
4 | sseq1 3115 | . . . . 5 | |
5 | 4 | anbi2d 459 | . . . 4 |
6 | imaeq2 4872 | . . . . . 6 | |
7 | 6 | eleq2d 2207 | . . . . 5 |
8 | 7 | notbid 656 | . . . . 5 |
9 | 7, 8 | orbi12d 782 | . . . 4 |
10 | 5, 9 | imbi12d 233 | . . 3 |
11 | sseq1 3115 | . . . . 5 | |
12 | 11 | anbi2d 459 | . . . 4 |
13 | imaeq2 4872 | . . . . . 6 | |
14 | 13 | eleq2d 2207 | . . . . 5 |
15 | 14 | notbid 656 | . . . . 5 |
16 | 14, 15 | orbi12d 782 | . . . 4 |
17 | 12, 16 | imbi12d 233 | . . 3 |
18 | sseq1 3115 | . . . . 5 | |
19 | 18 | anbi2d 459 | . . . 4 |
20 | imaeq2 4872 | . . . . . 6 | |
21 | 20 | eleq2d 2207 | . . . . 5 |
22 | 21 | notbid 656 | . . . . 5 |
23 | 21, 22 | orbi12d 782 | . . . 4 |
24 | 19, 23 | imbi12d 233 | . . 3 |
25 | sseq1 3115 | . . . . 5 | |
26 | 25 | anbi2d 459 | . . . 4 |
27 | imaeq2 4872 | . . . . . 6 | |
28 | 27 | eleq2d 2207 | . . . . 5 |
29 | 28 | notbid 656 | . . . . 5 |
30 | 28, 29 | orbi12d 782 | . . . 4 |
31 | 26, 30 | imbi12d 233 | . . 3 |
32 | noel 3362 | . . . . . 6 | |
33 | ima0 4893 | . . . . . . 7 | |
34 | 33 | eleq2i 2204 | . . . . . 6 |
35 | 32, 34 | mtbir 660 | . . . . 5 |
36 | 35 | a1i 9 | . . . 4 |
37 | 36 | olcd 723 | . . 3 |
38 | fidcenumlemr.dc | . . . . . 6 DECID | |
39 | 38 | ad2antrl 481 | . . . . 5 DECID |
40 | fidcenumlemr.f | . . . . . 6 | |
41 | 40 | ad2antrl 481 | . . . . 5 |
42 | simpll 518 | . . . . 5 | |
43 | simprr 521 | . . . . 5 | |
44 | simprl 520 | . . . . . 6 | |
45 | sssucid 4332 | . . . . . . 7 | |
46 | 45, 43 | sstrid 3103 | . . . . . 6 |
47 | simplr 519 | . . . . . 6 | |
48 | 44, 46, 47 | mp2and 429 | . . . . 5 |
49 | fidcenumlemrk.x | . . . . . 6 | |
50 | 49 | ad2antrl 481 | . . . . 5 |
51 | 39, 41, 42, 43, 48, 50 | fidcenumlemrks 6834 | . . . 4 |
52 | 51 | exp31 361 | . . 3 |
53 | 10, 17, 24, 31, 37, 52 | finds 4509 | . 2 |
54 | 1, 3, 53 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 wceq 1331 wcel 1480 wral 2414 wss 3066 c0 3358 csuc 4282 com 4499 cima 4537 wfo 5116 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fo 5124 df-fv 5126 |
This theorem is referenced by: fidcenumlemr 6836 ennnfonelemdc 11901 |
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