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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-charfunbi | Unicode version |
Description: In an ambient set , if membership in is stable, then it is
decidable if and only if has a characteristic function.
This characterization can be applied to singletons when the set has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.) |
Ref | Expression |
---|---|
bj-charfunbi.ex | |
bj-charfunbi.st | STAB |
Ref | Expression |
---|---|
bj-charfunbi | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2218 | . . . . 5 | |
2 | 1 | dcbid 824 | . . . 4 DECID DECID |
3 | 2 | cbvralvw 2684 | . . 3 DECID DECID |
4 | eleq1w 2218 | . . . . . . . . . . . 12 | |
5 | 4 | ifbid 3526 | . . . . . . . . . . 11 |
6 | 5 | cbvmptv 4060 | . . . . . . . . . 10 |
7 | 6 | a1i 9 | . . . . . . . . 9 DECID |
8 | 3 | biimpri 132 | . . . . . . . . . 10 DECID DECID |
9 | 8 | adantl 275 | . . . . . . . . 9 DECID DECID |
10 | 7, 9 | bj-charfundc 13343 | . . . . . . . 8 DECID |
11 | 10 | ex 114 | . . . . . . 7 DECID |
12 | 2on 6366 | . . . . . . . . . . 11 | |
13 | 12 | a1i 9 | . . . . . . . . . 10 |
14 | bj-charfunbi.ex | . . . . . . . . . 10 | |
15 | 13, 14 | elmapd 6600 | . . . . . . . . 9 |
16 | 15 | biimprd 157 | . . . . . . . 8 |
17 | 16 | adantrd 277 | . . . . . . 7 |
18 | 11, 17 | syld 45 | . . . . . 6 DECID |
19 | 18 | imp 123 | . . . . 5 DECID |
20 | fveq1 5464 | . . . . . . . . 9 | |
21 | 20 | eqeq1d 2166 | . . . . . . . 8 |
22 | 21 | ralbidv 2457 | . . . . . . 7 |
23 | 20 | eqeq1d 2166 | . . . . . . . 8 |
24 | 23 | ralbidv 2457 | . . . . . . 7 |
25 | 22, 24 | anbi12d 465 | . . . . . 6 |
26 | 25 | adantl 275 | . . . . 5 DECID |
27 | 10 | simprd 113 | . . . . 5 DECID |
28 | 19, 26, 27 | rspcedvd 2822 | . . . 4 DECID |
29 | 28 | ex 114 | . . 3 DECID |
30 | 3, 29 | syl5bi 151 | . 2 DECID |
31 | omex 4550 | . . . . . . . . 9 | |
32 | 2ssom 13337 | . . . . . . . . 9 | |
33 | mapss 6629 | . . . . . . . . 9 | |
34 | 31, 32, 33 | mp2an 423 | . . . . . . . 8 |
35 | fveq1 5464 | . . . . . . . . . . . . 13 | |
36 | 35 | eqeq1d 2166 | . . . . . . . . . . . 12 |
37 | 36 | ralbidv 2457 | . . . . . . . . . . 11 |
38 | 35 | eqeq1d 2166 | . . . . . . . . . . . 12 |
39 | 38 | ralbidv 2457 | . . . . . . . . . . 11 |
40 | 37, 39 | anbi12d 465 | . . . . . . . . . 10 |
41 | 40 | cbvrexvw 2685 | . . . . . . . . 9 |
42 | fveqeq2 5474 | . . . . . . . . . . . . 13 | |
43 | 42 | cbvralvw 2684 | . . . . . . . . . . . 12 |
44 | 1n0 6373 | . . . . . . . . . . . . . . . 16 | |
45 | 44 | neii 2329 | . . . . . . . . . . . . . . 15 |
46 | eqeq1 2164 | . . . . . . . . . . . . . . 15 | |
47 | 45, 46 | mtbiri 665 | . . . . . . . . . . . . . 14 |
48 | 47 | neqned 2334 | . . . . . . . . . . . . 13 |
49 | 48 | ralimi 2520 | . . . . . . . . . . . 12 |
50 | 43, 49 | sylbi 120 | . . . . . . . . . . 11 |
51 | fveqeq2 5474 | . . . . . . . . . . . . 13 | |
52 | 51 | cbvralvw 2684 | . . . . . . . . . . . 12 |
53 | 52 | biimpi 119 | . . . . . . . . . . 11 |
54 | 50, 53 | anim12i 336 | . . . . . . . . . 10 |
55 | 54 | reximi 2554 | . . . . . . . . 9 |
56 | 41, 55 | sylbi 120 | . . . . . . . 8 |
57 | ssrexv 3193 | . . . . . . . 8 | |
58 | 34, 56, 57 | mpsyl 65 | . . . . . . 7 |
59 | 58 | adantl 275 | . . . . . 6 |
60 | 59 | bj-charfunr 13345 | . . . . 5 DECID |
61 | 60 | ex 114 | . . . 4 DECID |
62 | eleq1w 2218 | . . . . . . 7 | |
63 | 62 | notbid 657 | . . . . . 6 |
64 | 63 | dcbid 824 | . . . . 5 DECID DECID |
65 | 64 | cbvralvw 2684 | . . . 4 DECID DECID |
66 | 61, 65 | syl6ibr 161 | . . 3 DECID |
67 | bj-charfunbi.st | . . . . . 6 STAB | |
68 | 67 | r19.21bi 2545 | . . . . 5 STAB |
69 | stdcn 833 | . . . . 5 STAB DECID DECID | |
70 | 68, 69 | sylib 121 | . . . 4 DECID DECID |
71 | 70 | ralimdva 2524 | . . 3 DECID DECID |
72 | 66, 71 | syld 45 | . 2 DECID |
73 | 30, 72 | impbid 128 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 STAB wstab 816 DECID wdc 820 wceq 1335 wcel 2128 wne 2327 wral 2435 wrex 2436 cvv 2712 cdif 3099 cin 3101 wss 3102 c0 3394 cif 3505 cmpt 4025 con0 4322 com 4547 wf 5163 cfv 5167 (class class class)co 5818 c1o 6350 c2o 6351 cmap 6586 |
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 4134 ax-pr 4168 ax-un 4392 ax-setind 4494 ax-iinf 4545 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 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-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 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-mpt 4027 df-tr 4063 df-id 4252 df-iord 4325 df-on 4327 df-suc 4330 df-iom 4548 df-xp 4589 df-rel 4590 df-cnv 4591 df-co 4592 df-dm 4593 df-rn 4594 df-res 4595 df-ima 4596 df-iota 5132 df-fun 5169 df-fn 5170 df-f 5171 df-fv 5175 df-ov 5821 df-oprab 5822 df-mpo 5823 df-1o 6357 df-2o 6358 df-map 6588 |
This theorem is referenced by: (None) |
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