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Mirrors > Home > ILE Home > Th. List > unsnfidcel | Unicode version |
Description: The condition in unsnfi 6807. This is intended to show that unsnfi 6807 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 6655 | . . . . 5 | |
2 | 1 | biimpi 119 | . . . 4 |
3 | 2 | 3ad2ant1 1002 | . . 3 |
4 | isfi 6655 | . . . . . . 7 | |
5 | 4 | biimpi 119 | . . . . . 6 |
6 | 5 | 3ad2ant3 1004 | . . . . 5 |
7 | 6 | adantr 274 | . . . 4 |
8 | simprr 521 | . . . . . . . . . 10 | |
9 | 8 | ad2antrr 479 | . . . . . . . . 9 |
10 | simprr 521 | . . . . . . . . . . . 12 | |
11 | 10 | ad3antrrr 483 | . . . . . . . . . . 11 |
12 | simplr 519 | . . . . . . . . . . 11 | |
13 | 11, 12 | breqtrrd 3956 | . . . . . . . . . 10 |
14 | 13 | ensymd 6677 | . . . . . . . . 9 |
15 | entr 6678 | . . . . . . . . 9 | |
16 | 9, 14, 15 | syl2anc 408 | . . . . . . . 8 |
17 | 16 | ensymd 6677 | . . . . . . 7 |
18 | simp1 981 | . . . . . . . . 9 | |
19 | 18 | ad4antr 485 | . . . . . . . 8 |
20 | simpl2 985 | . . . . . . . . . . 11 | |
21 | 20 | ad3antrrr 483 | . . . . . . . . . 10 |
22 | 21 | elexd 2699 | . . . . . . . . 9 |
23 | simpr 109 | . . . . . . . . 9 | |
24 | 22, 23 | eldifd 3081 | . . . . . . . 8 |
25 | php5fin 6776 | . . . . . . . 8 | |
26 | 19, 24, 25 | syl2anc 408 | . . . . . . 7 |
27 | 17, 26 | pm2.65da 650 | . . . . . 6 |
28 | 27 | olcd 723 | . . . . 5 |
29 | 8 | ad2antrr 479 | . . . . . . . . . . 11 |
30 | snssi 3664 | . . . . . . . . . . . . . 14 | |
31 | ssequn2 3249 | . . . . . . . . . . . . . 14 | |
32 | 30, 31 | sylib 121 | . . . . . . . . . . . . 13 |
33 | 32 | breq1d 3939 | . . . . . . . . . . . 12 |
34 | 33 | adantl 275 | . . . . . . . . . . 11 |
35 | 29, 34 | mpbid 146 | . . . . . . . . . 10 |
36 | 35 | ensymd 6677 | . . . . . . . . 9 |
37 | 10 | ad3antrrr 483 | . . . . . . . . 9 |
38 | entr 6678 | . . . . . . . . 9 | |
39 | 36, 37, 38 | syl2anc 408 | . . . . . . . 8 |
40 | simprl 520 | . . . . . . . . . 10 | |
41 | 40 | ad2antrr 479 | . . . . . . . . 9 |
42 | simprl 520 | . . . . . . . . . 10 | |
43 | 42 | ad3antrrr 483 | . . . . . . . . 9 |
44 | nneneq 6751 | . . . . . . . . 9 | |
45 | 41, 43, 44 | syl2anc 408 | . . . . . . . 8 |
46 | 39, 45 | mpbid 146 | . . . . . . 7 |
47 | simplr 519 | . . . . . . 7 | |
48 | 46, 47 | pm2.65da 650 | . . . . . 6 |
49 | 48 | orcd 722 | . . . . 5 |
50 | 42 | adantr 274 | . . . . . . 7 |
51 | nndceq 6395 | . . . . . . 7 DECID | |
52 | 40, 50, 51 | syl2anc 408 | . . . . . 6 DECID |
53 | exmiddc 821 | . . . . . 6 DECID | |
54 | 52, 53 | syl 14 | . . . . 5 |
55 | 28, 49, 54 | mpjaodan 787 | . . . 4 |
56 | 7, 55 | rexlimddv 2554 | . . 3 |
57 | 3, 56 | rexlimddv 2554 | . 2 |
58 | df-dc 820 | . 2 DECID | |
59 | 57, 58 | sylibr 133 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 w3a 962 wceq 1331 wcel 1480 wrex 2417 cvv 2686 cdif 3068 cun 3069 wss 3071 csn 3527 class class class wbr 3929 com 4504 cen 6632 cfn 6634 |
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-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-3or 963 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-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 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-1o 6313 df-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: (None) |
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