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Mirrors > Home > ILE Home > Th. List > unsnfidcel | Unicode version |
Description: The condition in unsnfi 6384. This is intended to show that unsnfi 6384 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.) |
Ref | Expression |
---|---|
unsnfidcel | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6264 | . . . . 5 | |
2 | 1 | biimpi 118 | . . . 4 |
3 | 2 | 3ad2ant1 959 | . . 3 |
4 | isfi 6264 | . . . . . . 7 | |
5 | 4 | biimpi 118 | . . . . . 6 |
6 | 5 | 3ad2ant3 961 | . . . . 5 |
7 | 6 | adantr 270 | . . . 4 |
8 | simprr 498 | . . . . . . . . . 10 | |
9 | 8 | ad2antrr 471 | . . . . . . . . 9 |
10 | simprr 498 | . . . . . . . . . . . 12 | |
11 | 10 | ad3antrrr 475 | . . . . . . . . . . 11 |
12 | simplr 496 | . . . . . . . . . . 11 | |
13 | 11, 12 | breqtrrd 3811 | . . . . . . . . . 10 |
14 | 13 | ensymd 6286 | . . . . . . . . 9 |
15 | entr 6287 | . . . . . . . . 9 | |
16 | 9, 14, 15 | syl2anc 403 | . . . . . . . 8 |
17 | 16 | ensymd 6286 | . . . . . . 7 |
18 | simp1 938 | . . . . . . . . 9 | |
19 | 18 | ad4antr 477 | . . . . . . . 8 |
20 | simpl2 942 | . . . . . . . . . . 11 | |
21 | 20 | ad3antrrr 475 | . . . . . . . . . 10 |
22 | 21 | elexd 2612 | . . . . . . . . 9 |
23 | simpr 108 | . . . . . . . . 9 | |
24 | 22, 23 | eldifd 2983 | . . . . . . . 8 |
25 | php5fin 6366 | . . . . . . . 8 | |
26 | 19, 24, 25 | syl2anc 403 | . . . . . . 7 |
27 | 17, 26 | pm2.65da 619 | . . . . . 6 |
28 | 27 | olcd 685 | . . . . 5 |
29 | 8 | ad2antrr 471 | . . . . . . . . . . 11 |
30 | snssi 3529 | . . . . . . . . . . . . . 14 | |
31 | ssequn2 3145 | . . . . . . . . . . . . . 14 | |
32 | 30, 31 | sylib 120 | . . . . . . . . . . . . 13 |
33 | 32 | breq1d 3795 | . . . . . . . . . . . 12 |
34 | 33 | adantl 271 | . . . . . . . . . . 11 |
35 | 29, 34 | mpbid 145 | . . . . . . . . . 10 |
36 | 35 | ensymd 6286 | . . . . . . . . 9 |
37 | 10 | ad3antrrr 475 | . . . . . . . . 9 |
38 | entr 6287 | . . . . . . . . 9 | |
39 | 36, 37, 38 | syl2anc 403 | . . . . . . . 8 |
40 | simprl 497 | . . . . . . . . . 10 | |
41 | 40 | ad2antrr 471 | . . . . . . . . 9 |
42 | simprl 497 | . . . . . . . . . 10 | |
43 | 42 | ad3antrrr 475 | . . . . . . . . 9 |
44 | nneneq 6343 | . . . . . . . . 9 | |
45 | 41, 43, 44 | syl2anc 403 | . . . . . . . 8 |
46 | 39, 45 | mpbid 145 | . . . . . . 7 |
47 | simplr 496 | . . . . . . 7 | |
48 | 46, 47 | pm2.65da 619 | . . . . . 6 |
49 | 48 | orcd 684 | . . . . 5 |
50 | 42 | adantr 270 | . . . . . . 7 |
51 | nndceq 6100 | . . . . . . 7 DECID | |
52 | 40, 50, 51 | syl2anc 403 | . . . . . 6 DECID |
53 | exmiddc 777 | . . . . . 6 DECID | |
54 | 52, 53 | syl 14 | . . . . 5 |
55 | 28, 49, 54 | mpjaodan 744 | . . . 4 |
56 | 7, 55 | rexlimddv 2481 | . . 3 |
57 | 3, 56 | rexlimddv 2481 | . 2 |
58 | df-dc 776 | . 2 DECID | |
59 | 57, 58 | sylibr 132 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 DECID wdc 775 w3a 919 wceq 1284 wcel 1433 wrex 2349 cvv 2601 cdif 2970 cun 2971 wss 2973 csn 3398 class class class wbr 3785 com 4331 cen 6242 cfn 6244 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-1o 6024 df-er 6129 df-en 6245 df-fin 6247 |
This theorem is referenced by: (None) |
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