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Mirrors > Home > ILE Home > Th. List > Mathboxes > pw1nct | Unicode version |
Description: A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.) |
Ref | Expression |
---|---|
pw1nct | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1515 | . . . . . . . 8 | |
2 | nfv 1515 | . . . . . . . . 9 | |
3 | nfre1 2507 | . . . . . . . . 9 | |
4 | 2, 3 | nfim 1559 | . . . . . . . 8 |
5 | 1, 4 | nfim 1559 | . . . . . . 7 |
6 | 5 | nfal 1563 | . . . . . 6 |
7 | nfv 1515 | . . . . . 6 | |
8 | 6, 7 | nfan 1552 | . . . . 5 |
9 | breq1 3979 | . . . . . . . . 9 | |
10 | simpr 109 | . . . . . . . . 9 | |
11 | 0elpw 4137 | . . . . . . . . . 10 | |
12 | 11 | a1i 9 | . . . . . . . . 9 |
13 | 9, 10, 12 | rspcdva 2830 | . . . . . . . 8 |
14 | 0ex 4103 | . . . . . . . . 9 | |
15 | vex 2724 | . . . . . . . . 9 | |
16 | 14, 15 | brcnv 4781 | . . . . . . . 8 |
17 | 13, 16 | sylib 121 | . . . . . . 7 |
18 | fofn 5406 | . . . . . . . . 9 | |
19 | 18 | ad3antlr 485 | . . . . . . . 8 |
20 | simplr 520 | . . . . . . . 8 | |
21 | fnbrfvb 5521 | . . . . . . . 8 | |
22 | 19, 20, 21 | syl2anc 409 | . . . . . . 7 |
23 | 17, 22 | mpbird 166 | . . . . . 6 |
24 | breq1 3979 | . . . . . . . . . 10 | |
25 | 1oex 6383 | . . . . . . . . . . . 12 | |
26 | 25 | pwid 3568 | . . . . . . . . . . 11 |
27 | 26 | a1i 9 | . . . . . . . . . 10 |
28 | 24, 10, 27 | rspcdva 2830 | . . . . . . . . 9 |
29 | 25, 15 | brcnv 4781 | . . . . . . . . 9 |
30 | 28, 29 | sylib 121 | . . . . . . . 8 |
31 | fnbrfvb 5521 | . . . . . . . . 9 | |
32 | 19, 20, 31 | syl2anc 409 | . . . . . . . 8 |
33 | 30, 32 | mpbird 166 | . . . . . . 7 |
34 | 1n0 6391 | . . . . . . . 8 | |
35 | 34 | neii 2336 | . . . . . . 7 |
36 | eqeq1 2171 | . . . . . . . . 9 | |
37 | 36 | biimpd 143 | . . . . . . . 8 |
38 | 37 | con3dimp 625 | . . . . . . 7 |
39 | 33, 35, 38 | sylancl 410 | . . . . . 6 |
40 | 23, 39 | pm2.21fal 1362 | . . . . 5 |
41 | fof 5404 | . . . . . . . 8 | |
42 | fssxp 5349 | . . . . . . . . . 10 | |
43 | cnvss 4771 | . . . . . . . . . 10 | |
44 | 42, 43 | syl 14 | . . . . . . . . 9 |
45 | cnvxp 5016 | . . . . . . . . 9 | |
46 | 44, 45 | sseqtrdi 3185 | . . . . . . . 8 |
47 | 41, 46 | syl 14 | . . . . . . 7 |
48 | 47 | adantl 275 | . . . . . 6 |
49 | foelrn 5715 | . . . . . . . . 9 | |
50 | 18 | ad2antrr 480 | . . . . . . . . . . 11 |
51 | simpr 109 | . . . . . . . . . . 11 | |
52 | eqcom 2166 | . . . . . . . . . . . 12 | |
53 | fnbrfvb 5521 | . . . . . . . . . . . . 13 | |
54 | brcnvg 4779 | . . . . . . . . . . . . . . 15 | |
55 | 54 | elvd 2726 | . . . . . . . . . . . . . 14 |
56 | 55 | elv 2725 | . . . . . . . . . . . . 13 |
57 | 53, 56 | bitr4di 197 | . . . . . . . . . . . 12 |
58 | 52, 57 | bitr3id 193 | . . . . . . . . . . 11 |
59 | 50, 51, 58 | syl2anc 409 | . . . . . . . . . 10 |
60 | 59 | rexbidva 2461 | . . . . . . . . 9 |
61 | 49, 60 | mpbid 146 | . . . . . . . 8 |
62 | 61 | ralrimiva 2537 | . . . . . . 7 |
63 | 62 | adantl 275 | . . . . . 6 |
64 | cnvexg 5135 | . . . . . . . 8 | |
65 | 64 | elv 2725 | . . . . . . 7 |
66 | simpl 108 | . . . . . . 7 | |
67 | sseq1 3160 | . . . . . . . . 9 | |
68 | breq 3978 | . . . . . . . . . . . 12 | |
69 | 68 | rexbidv 2465 | . . . . . . . . . . 11 |
70 | 69 | ralbidv 2464 | . . . . . . . . . 10 |
71 | breq 3978 | . . . . . . . . . . . 12 | |
72 | 71 | ralbidv 2464 | . . . . . . . . . . 11 |
73 | 72 | rexbidv 2465 | . . . . . . . . . 10 |
74 | 70, 73 | imbi12d 233 | . . . . . . . . 9 |
75 | 67, 74 | imbi12d 233 | . . . . . . . 8 |
76 | 75 | spcgv 2808 | . . . . . . 7 |
77 | 65, 66, 76 | mpsyl 65 | . . . . . 6 |
78 | 48, 63, 77 | mp2d 47 | . . . . 5 |
79 | 8, 40, 78 | r19.29af 2605 | . . . 4 |
80 | 79 | inegd 1361 | . . 3 |
81 | 80 | nexdv 1923 | . 2 |
82 | elex2 2737 | . . 3 | |
83 | ctm 7065 | . . 3 ⊔ | |
84 | 11, 82, 83 | mp2b 8 | . 2 ⊔ |
85 | 81, 84 | sylnibr 667 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wal 1340 wceq 1342 wfal 1347 wex 1479 wcel 2135 wral 2442 wrex 2443 cvv 2721 wss 3111 c0 3404 cpw 3553 class class class wbr 3976 com 4561 cxp 4596 ccnv 4597 wfn 5177 wf 5178 wfo 5180 cfv 5182 c1o 6368 ⊔ cdju 6993 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-1st 6100 df-2nd 6101 df-1o 6375 df-dju 6994 df-inl 7003 df-inr 7004 df-case 7040 |
This theorem is referenced by: (None) |
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