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Mirrors > Home > ILE Home > Th. List > exmidsssnc | Unicode version |
Description: Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4121 but lets you choose any set as the element of the singleton rather than just . It is similar to exmidsssn 4125 but for a particular set rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.) |
Ref | Expression |
---|---|
exmidsssnc | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidsssn 4125 | . . . 4 EXMID | |
2 | sneq 3538 | . . . . . . . 8 | |
3 | 2 | sseq2d 3127 | . . . . . . 7 |
4 | 2 | eqeq2d 2151 | . . . . . . . 8 |
5 | 4 | orbi2d 779 | . . . . . . 7 |
6 | 3, 5 | bibi12d 234 | . . . . . 6 |
7 | 6 | spcgv 2773 | . . . . 5 |
8 | 7 | alimdv 1851 | . . . 4 |
9 | 1, 8 | syl5bi 151 | . . 3 EXMID |
10 | bi1 117 | . . . 4 | |
11 | 10 | alimi 1431 | . . 3 |
12 | 9, 11 | syl6 33 | . 2 EXMID |
13 | ssrab2 3182 | . . . . . . . . 9 | |
14 | snexg 4108 | . . . . . . . . . 10 | |
15 | rabexg 4071 | . . . . . . . . . 10 | |
16 | sseq1 3120 | . . . . . . . . . . . 12 | |
17 | eqeq1 2146 | . . . . . . . . . . . . 13 | |
18 | eqeq1 2146 | . . . . . . . . . . . . 13 | |
19 | 17, 18 | orbi12d 782 | . . . . . . . . . . . 12 |
20 | 16, 19 | imbi12d 233 | . . . . . . . . . . 11 |
21 | 20 | spcgv 2773 | . . . . . . . . . 10 |
22 | 14, 15, 21 | 3syl 17 | . . . . . . . . 9 |
23 | 13, 22 | mpii 44 | . . . . . . . 8 |
24 | snmg 3641 | . . . . . . . . . . . 12 | |
25 | r19.3rmv 3453 | . . . . . . . . . . . 12 | |
26 | 24, 25 | syl 14 | . . . . . . . . . . 11 |
27 | rabeq0 3392 | . . . . . . . . . . 11 | |
28 | 26, 27 | syl6rbbr 198 | . . . . . . . . . 10 |
29 | 28 | biimpd 143 | . . . . . . . . 9 |
30 | snidg 3554 | . . . . . . . . . . . . 13 | |
31 | 30 | adantr 274 | . . . . . . . . . . . 12 |
32 | simpr 109 | . . . . . . . . . . . 12 | |
33 | 31, 32 | eleqtrrd 2219 | . . . . . . . . . . 11 |
34 | biidd 171 | . . . . . . . . . . . . 13 | |
35 | 34 | elrab 2840 | . . . . . . . . . . . 12 |
36 | 35 | simprbi 273 | . . . . . . . . . . 11 |
37 | 33, 36 | syl 14 | . . . . . . . . . 10 |
38 | 37 | ex 114 | . . . . . . . . 9 |
39 | 29, 38 | orim12d 775 | . . . . . . . 8 |
40 | 23, 39 | syld 45 | . . . . . . 7 |
41 | orcom 717 | . . . . . . 7 | |
42 | 40, 41 | syl6ib 160 | . . . . . 6 |
43 | df-dc 820 | . . . . . 6 DECID | |
44 | 42, 43 | syl6ibr 161 | . . . . 5 DECID |
45 | 44 | a1dd 48 | . . . 4 DECID |
46 | 45 | alrimdv 1848 | . . 3 DECID |
47 | df-exmid 4119 | . . 3 EXMID DECID | |
48 | 46, 47 | syl6ibr 161 | . 2 EXMID |
49 | 12, 48 | impbid 128 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 wal 1329 wceq 1331 wex 1468 wcel 1480 wral 2416 crab 2420 cvv 2686 wss 3071 c0 3363 csn 3527 EXMIDwem 4118 |
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-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 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rab 2425 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-exmid 4119 |
This theorem is referenced by: exmidunben 11939 |
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