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Mirrors > Home > ILE Home > Th. List > ctiunct | Unicode version |
Description: A sequence of
enumerations gives an enumeration of the union. We refer
to "sequence of enumerations" rather than "countably many
countable
sets" because the hypothesis provides more than countability for
each
: it refers to together with the
which enumerates it. Theorem 8.1.19 of [AczelRathjen], p. 74.
For "countably many countable sets" the key hypothesis would be ⊔ . This is almost omiunct 12320 (which uses countable choice) although that is for a countably infinite collection not any countable collection. Compare with the case of two sets instead of countably many, as seen at unct 12318, which says that the union of two countable sets is countable . The proof proceeds by mapping a natural number to a pair of natural numbers (by xpomen 12271) and using the first number to map to an element of and the second number to map to an element of B(x) . In this way we are able to map to every element of . Although it would be possible to work directly with countability expressed as ⊔ , we instead use functions from subsets of the natural numbers via ctssdccl 7067 and ctssdc 7069. (Contributed by Jim Kingdon, 31-Oct-2023.) |
Ref | Expression |
---|---|
ctiunct.a | ⊔ |
ctiunct.b | ⊔ |
Ref | Expression |
---|---|
ctiunct | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpomen 12271 | . . . . 5 | |
2 | 1 | ensymi 6739 | . . . 4 |
3 | bren 6704 | . . . 4 | |
4 | 2, 3 | mpbi 144 | . . 3 |
5 | 4 | a1i 9 | . 2 |
6 | ctiunct.a | . . . . . . . 8 ⊔ | |
7 | eqid 2164 | . . . . . . . 8 inl inl | |
8 | eqid 2164 | . . . . . . . 8 inl inl | |
9 | 6, 7, 8 | ctssdccl 7067 | . . . . . . 7 inl inl inl DECID inl |
10 | 9 | simp1d 998 | . . . . . 6 inl |
11 | 10 | adantr 274 | . . . . 5 inl |
12 | 9 | simp3d 1000 | . . . . . 6 DECID inl |
13 | 12 | adantr 274 | . . . . 5 DECID inl |
14 | 9 | simp2d 999 | . . . . . 6 inl inl |
15 | 14 | adantr 274 | . . . . 5 inl inl |
16 | ctiunct.b | . . . . . . . 8 ⊔ | |
17 | eqid 2164 | . . . . . . . 8 inl inl | |
18 | eqid 2164 | . . . . . . . 8 inl inl | |
19 | 16, 17, 18 | ctssdccl 7067 | . . . . . . 7 inl inl inl DECID inl |
20 | 19 | simp1d 998 | . . . . . 6 inl |
21 | 20 | adantlr 469 | . . . . 5 inl |
22 | 19 | simp3d 1000 | . . . . . 6 DECID inl |
23 | 22 | adantlr 469 | . . . . 5 DECID inl |
24 | 19 | simp2d 999 | . . . . . 6 inl inl |
25 | 24 | adantlr 469 | . . . . 5 inl inl |
26 | simpr 109 | . . . . 5 | |
27 | eqid 2164 | . . . . 5 inl inl inl inl inl inl | |
28 | 11, 13, 15, 21, 23, 25, 26, 27 | ctiunctlemuom 12312 | . . . 4 inl inl inl |
29 | eqid 2164 | . . . . . 6 inl inl inl inl inl inl inl inl inl inl | |
30 | nfv 1515 | . . . . . . . . 9 inl | |
31 | nfcsb1v 3073 | . . . . . . . . . 10 inl inl | |
32 | 31 | nfel2 2319 | . . . . . . . . 9 inl inl |
33 | 30, 32 | nfan 1552 | . . . . . . . 8 inl inl inl |
34 | nfcv 2306 | . . . . . . . 8 | |
35 | 33, 34 | nfrabxy 2644 | . . . . . . 7 inl inl inl |
36 | nfcsb1v 3073 | . . . . . . . 8 inl inl | |
37 | nfcv 2306 | . . . . . . . 8 | |
38 | 36, 37 | nffv 5490 | . . . . . . 7 inl inl |
39 | 35, 38 | nfmpt 4068 | . . . . . 6 inl inl inl inl inl |
40 | 11, 13, 15, 21, 23, 25, 26, 27, 29, 39, 35 | ctiunctlemfo 12315 | . . . . 5 inl inl inl inl inl inl inl inl |
41 | omex 4564 | . . . . . . . 8 | |
42 | 41 | rabex 4120 | . . . . . . 7 inl inl inl |
43 | 42 | mptex 5705 | . . . . . 6 inl inl inl inl inl |
44 | foeq1 5400 | . . . . . 6 inl inl inl inl inl inl inl inl inl inl inl inl inl inl inl inl | |
45 | 43, 44 | spcev 2816 | . . . . 5 inl inl inl inl inl inl inl inl inl inl inl |
46 | 40, 45 | syl 14 | . . . 4 inl inl inl |
47 | 11, 13, 15, 21, 23, 25, 26, 27 | ctiunctlemudc 12313 | . . . 4 DECID inl inl inl |
48 | sseq1 3160 | . . . . . 6 inl inl inl inl inl inl | |
49 | foeq2 5401 | . . . . . . 7 inl inl inl inl inl inl | |
50 | 49 | exbidv 1812 | . . . . . 6 inl inl inl inl inl inl |
51 | eleq2 2228 | . . . . . . . 8 inl inl inl inl inl inl | |
52 | 51 | dcbid 828 | . . . . . . 7 inl inl inl DECID DECID inl inl inl |
53 | 52 | ralbidv 2464 | . . . . . 6 inl inl inl DECID DECID inl inl inl |
54 | 48, 50, 53 | 3anbi123d 1301 | . . . . 5 inl inl inl DECID inl inl inl inl inl inl DECID inl inl inl |
55 | 42, 54 | spcev 2816 | . . . 4 inl inl inl inl inl inl DECID inl inl inl DECID |
56 | 28, 46, 47, 55 | syl3anc 1227 | . . 3 DECID |
57 | ctssdc 7069 | . . . 4 DECID ⊔ | |
58 | foeq1 5400 | . . . . 5 ⊔ ⊔ | |
59 | 58 | cbvexv 1905 | . . . 4 ⊔ ⊔ |
60 | 57, 59 | bitri 183 | . . 3 DECID ⊔ |
61 | 56, 60 | sylib 121 | . 2 ⊔ |
62 | 5, 61 | exlimddv 1885 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 824 w3a 967 wceq 1342 wex 1479 wcel 2135 wral 2442 crab 2446 csb 3040 wss 3111 ciun 3860 class class class wbr 3976 cmpt 4037 com 4561 cxp 4596 ccnv 4597 cima 4601 ccom 4602 wfo 5180 wf1o 5181 cfv 5182 c1st 6098 c2nd 6099 c1o 6368 cen 6695 ⊔ cdju 6993 inlcinl 7001 |
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-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-xor 1365 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-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 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-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 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-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-1o 6375 df-er 6492 df-en 6698 df-dju 6994 df-inl 7003 df-inr 7004 df-case 7040 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-dvds 11714 |
This theorem is referenced by: ctiunctal 12317 unct 12318 |
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