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Mirrors > Home > ILE Home > Th. List > unsnfidcel | Unicode version |
Description: The condition in unsnfi 6863. This is intended to show that unsnfi 6863 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 6706 | . . . . 5 | |
2 | 1 | biimpi 119 | . . . 4 |
3 | 2 | 3ad2ant1 1003 | . . 3 |
4 | isfi 6706 | . . . . . . 7 | |
5 | 4 | biimpi 119 | . . . . . 6 |
6 | 5 | 3ad2ant3 1005 | . . . . 5 |
7 | 6 | adantr 274 | . . . 4 |
8 | simprr 522 | . . . . . . . . . 10 | |
9 | 8 | ad2antrr 480 | . . . . . . . . 9 |
10 | simprr 522 | . . . . . . . . . . . 12 | |
11 | 10 | ad3antrrr 484 | . . . . . . . . . . 11 |
12 | simplr 520 | . . . . . . . . . . 11 | |
13 | 11, 12 | breqtrrd 3992 | . . . . . . . . . 10 |
14 | 13 | ensymd 6728 | . . . . . . . . 9 |
15 | entr 6729 | . . . . . . . . 9 | |
16 | 9, 14, 15 | syl2anc 409 | . . . . . . . 8 |
17 | 16 | ensymd 6728 | . . . . . . 7 |
18 | simp1 982 | . . . . . . . . 9 | |
19 | 18 | ad4antr 486 | . . . . . . . 8 |
20 | simpl2 986 | . . . . . . . . . . 11 | |
21 | 20 | ad3antrrr 484 | . . . . . . . . . 10 |
22 | 21 | elexd 2725 | . . . . . . . . 9 |
23 | simpr 109 | . . . . . . . . 9 | |
24 | 22, 23 | eldifd 3112 | . . . . . . . 8 |
25 | php5fin 6827 | . . . . . . . 8 | |
26 | 19, 24, 25 | syl2anc 409 | . . . . . . 7 |
27 | 17, 26 | pm2.65da 651 | . . . . . 6 |
28 | 27 | olcd 724 | . . . . 5 |
29 | 8 | ad2antrr 480 | . . . . . . . . . . 11 |
30 | snssi 3700 | . . . . . . . . . . . . . 14 | |
31 | ssequn2 3280 | . . . . . . . . . . . . . 14 | |
32 | 30, 31 | sylib 121 | . . . . . . . . . . . . 13 |
33 | 32 | breq1d 3975 | . . . . . . . . . . . 12 |
34 | 33 | adantl 275 | . . . . . . . . . . 11 |
35 | 29, 34 | mpbid 146 | . . . . . . . . . 10 |
36 | 35 | ensymd 6728 | . . . . . . . . 9 |
37 | 10 | ad3antrrr 484 | . . . . . . . . 9 |
38 | entr 6729 | . . . . . . . . 9 | |
39 | 36, 37, 38 | syl2anc 409 | . . . . . . . 8 |
40 | simprl 521 | . . . . . . . . . 10 | |
41 | 40 | ad2antrr 480 | . . . . . . . . 9 |
42 | simprl 521 | . . . . . . . . . 10 | |
43 | 42 | ad3antrrr 484 | . . . . . . . . 9 |
44 | nneneq 6802 | . . . . . . . . 9 | |
45 | 41, 43, 44 | syl2anc 409 | . . . . . . . 8 |
46 | 39, 45 | mpbid 146 | . . . . . . 7 |
47 | simplr 520 | . . . . . . 7 | |
48 | 46, 47 | pm2.65da 651 | . . . . . 6 |
49 | 48 | orcd 723 | . . . . 5 |
50 | 42 | adantr 274 | . . . . . . 7 |
51 | nndceq 6446 | . . . . . . 7 DECID | |
52 | 40, 50, 51 | syl2anc 409 | . . . . . 6 DECID |
53 | exmiddc 822 | . . . . . 6 DECID | |
54 | 52, 53 | syl 14 | . . . . 5 |
55 | 28, 49, 54 | mpjaodan 788 | . . . 4 |
56 | 7, 55 | rexlimddv 2579 | . . 3 |
57 | 3, 56 | rexlimddv 2579 | . 2 |
58 | df-dc 821 | . 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 698 DECID wdc 820 w3a 963 wceq 1335 wcel 2128 wrex 2436 cvv 2712 cdif 3099 cun 3100 wss 3102 csn 3560 class class class wbr 3965 com 4549 cen 6683 cfn 6685 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-1o 6363 df-er 6480 df-en 6686 df-fin 6688 |
This theorem is referenced by: (None) |
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