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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 2231 | . . . . 5 | |
2 | 1 | dcbid 833 | . . . 4 DECID DECID |
3 | 2 | cbvralvw 2700 | . . 3 DECID DECID |
4 | eleq1w 2231 | . . . . . . . . . . . 12 | |
5 | 4 | ifbid 3547 | . . . . . . . . . . 11 |
6 | 5 | cbvmptv 4085 | . . . . . . . . . 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 13843 | . . . . . . . 8 DECID |
11 | 10 | ex 114 | . . . . . . 7 DECID |
12 | 2on 6404 | . . . . . . . . . . 11 | |
13 | 12 | a1i 9 | . . . . . . . . . 10 |
14 | bj-charfunbi.ex | . . . . . . . . . 10 | |
15 | 13, 14 | elmapd 6640 | . . . . . . . . 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 5495 | . . . . . . . . 9 | |
21 | 20 | eqeq1d 2179 | . . . . . . . 8 |
22 | 21 | ralbidv 2470 | . . . . . . 7 |
23 | 20 | eqeq1d 2179 | . . . . . . . 8 |
24 | 23 | ralbidv 2470 | . . . . . . 7 |
25 | 22, 24 | anbi12d 470 | . . . . . 6 |
26 | 25 | adantl 275 | . . . . 5 DECID |
27 | 10 | simprd 113 | . . . . 5 DECID |
28 | 19, 26, 27 | rspcedvd 2840 | . . . 4 DECID |
29 | 28 | ex 114 | . . 3 DECID |
30 | 3, 29 | syl5bi 151 | . 2 DECID |
31 | omex 4577 | . . . . . . . . 9 | |
32 | 2ssom 6503 | . . . . . . . . 9 | |
33 | mapss 6669 | . . . . . . . . 9 | |
34 | 31, 32, 33 | mp2an 424 | . . . . . . . 8 |
35 | fveq1 5495 | . . . . . . . . . . . . 13 | |
36 | 35 | eqeq1d 2179 | . . . . . . . . . . . 12 |
37 | 36 | ralbidv 2470 | . . . . . . . . . . 11 |
38 | 35 | eqeq1d 2179 | . . . . . . . . . . . 12 |
39 | 38 | ralbidv 2470 | . . . . . . . . . . 11 |
40 | 37, 39 | anbi12d 470 | . . . . . . . . . 10 |
41 | 40 | cbvrexvw 2701 | . . . . . . . . 9 |
42 | fveqeq2 5505 | . . . . . . . . . . . . 13 | |
43 | 42 | cbvralvw 2700 | . . . . . . . . . . . 12 |
44 | 1n0 6411 | . . . . . . . . . . . . . . . 16 | |
45 | 44 | neii 2342 | . . . . . . . . . . . . . . 15 |
46 | eqeq1 2177 | . . . . . . . . . . . . . . 15 | |
47 | 45, 46 | mtbiri 670 | . . . . . . . . . . . . . 14 |
48 | 47 | neqned 2347 | . . . . . . . . . . . . 13 |
49 | 48 | ralimi 2533 | . . . . . . . . . . . 12 |
50 | 43, 49 | sylbi 120 | . . . . . . . . . . 11 |
51 | fveqeq2 5505 | . . . . . . . . . . . . 13 | |
52 | 51 | cbvralvw 2700 | . . . . . . . . . . . 12 |
53 | 52 | biimpi 119 | . . . . . . . . . . 11 |
54 | 50, 53 | anim12i 336 | . . . . . . . . . 10 |
55 | 54 | reximi 2567 | . . . . . . . . 9 |
56 | 41, 55 | sylbi 120 | . . . . . . . 8 |
57 | ssrexv 3212 | . . . . . . . 8 | |
58 | 34, 56, 57 | mpsyl 65 | . . . . . . 7 |
59 | 58 | adantl 275 | . . . . . 6 |
60 | 59 | bj-charfunr 13845 | . . . . 5 DECID |
61 | 60 | ex 114 | . . . 4 DECID |
62 | eleq1w 2231 | . . . . . . 7 | |
63 | 62 | notbid 662 | . . . . . 6 |
64 | 63 | dcbid 833 | . . . . 5 DECID DECID |
65 | 64 | cbvralvw 2700 | . . . 4 DECID DECID |
66 | 61, 65 | syl6ibr 161 | . . 3 DECID |
67 | bj-charfunbi.st | . . . . . 6 STAB | |
68 | 67 | r19.21bi 2558 | . . . . 5 STAB |
69 | stdcn 842 | . . . . 5 STAB DECID DECID | |
70 | 68, 69 | sylib 121 | . . . 4 DECID DECID |
71 | 70 | ralimdva 2537 | . . 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 825 DECID wdc 829 wceq 1348 wcel 2141 wne 2340 wral 2448 wrex 2449 cvv 2730 cdif 3118 cin 3120 wss 3121 c0 3414 cif 3526 cmpt 4050 con0 4348 com 4574 wf 5194 cfv 5198 (class class class)co 5853 c1o 6388 c2o 6389 cmap 6626 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1o 6395 df-2o 6396 df-map 6628 |
This theorem is referenced by: (None) |
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