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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.
The "countably many countable sets" version could be expressed as ⊔ and countable choice would be needed to derive the current hypothesis from that. Compare with the case of two sets instead of countably many, as seen at unct 11954, in which case we express countability using . The proof proceeds by mapping a natural number to a pair of natural numbers (by xpomen 11908) 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 6996 and ctssdc 6998. (Contributed by Jim Kingdon, 31-Oct-2023.) |
Ref | Expression |
---|---|
ctiunct.a | ⊔ |
ctiunct.b | ⊔ |
Ref | Expression |
---|---|
ctiunct | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpomen 11908 | . . . . 5 | |
2 | 1 | ensymi 6676 | . . . 4 |
3 | bren 6641 | . . . 4 | |
4 | 2, 3 | mpbi 144 | . . 3 |
5 | 4 | a1i 9 | . 2 |
6 | ctiunct.a | . . . . . . . 8 ⊔ | |
7 | eqid 2139 | . . . . . . . 8 inl inl | |
8 | eqid 2139 | . . . . . . . 8 inl inl | |
9 | 6, 7, 8 | ctssdccl 6996 | . . . . . . 7 inl inl inl DECID inl |
10 | 9 | simp1d 993 | . . . . . 6 inl |
11 | 10 | adantr 274 | . . . . 5 inl |
12 | 9 | simp3d 995 | . . . . . 6 DECID inl |
13 | 12 | adantr 274 | . . . . 5 DECID inl |
14 | 9 | simp2d 994 | . . . . . 6 inl inl |
15 | 14 | adantr 274 | . . . . 5 inl inl |
16 | ctiunct.b | . . . . . . . 8 ⊔ | |
17 | eqid 2139 | . . . . . . . 8 inl inl | |
18 | eqid 2139 | . . . . . . . 8 inl inl | |
19 | 16, 17, 18 | ctssdccl 6996 | . . . . . . 7 inl inl inl DECID inl |
20 | 19 | simp1d 993 | . . . . . 6 inl |
21 | 20 | adantlr 468 | . . . . 5 inl |
22 | 19 | simp3d 995 | . . . . . 6 DECID inl |
23 | 22 | adantlr 468 | . . . . 5 DECID inl |
24 | 19 | simp2d 994 | . . . . . 6 inl inl |
25 | 24 | adantlr 468 | . . . . 5 inl inl |
26 | simpr 109 | . . . . 5 | |
27 | eqid 2139 | . . . . 5 inl inl inl inl inl inl | |
28 | 11, 13, 15, 21, 23, 25, 26, 27 | ctiunctlemuom 11949 | . . . 4 inl inl inl |
29 | eqid 2139 | . . . . . 6 inl inl inl inl inl inl inl inl inl inl | |
30 | nfv 1508 | . . . . . . . . 9 inl | |
31 | nfcsb1v 3035 | . . . . . . . . . 10 inl inl | |
32 | 31 | nfel2 2294 | . . . . . . . . 9 inl inl |
33 | 30, 32 | nfan 1544 | . . . . . . . 8 inl inl inl |
34 | nfcv 2281 | . . . . . . . 8 | |
35 | 33, 34 | nfrabxy 2611 | . . . . . . 7 inl inl inl |
36 | nfcsb1v 3035 | . . . . . . . 8 inl inl | |
37 | nfcv 2281 | . . . . . . . 8 | |
38 | 36, 37 | nffv 5431 | . . . . . . 7 inl inl |
39 | 35, 38 | nfmpt 4020 | . . . . . 6 inl inl inl inl inl |
40 | 11, 13, 15, 21, 23, 25, 26, 27, 29, 39, 35 | ctiunctlemfo 11952 | . . . . 5 inl inl inl inl inl inl inl inl |
41 | omex 4507 | . . . . . . . 8 | |
42 | 41 | rabex 4072 | . . . . . . 7 inl inl inl |
43 | 42 | mptex 5646 | . . . . . 6 inl inl inl inl inl |
44 | foeq1 5341 | . . . . . 6 inl inl inl inl inl inl inl inl inl inl inl inl inl inl inl inl | |
45 | 43, 44 | spcev 2780 | . . . . 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 11950 | . . . 4 DECID inl inl inl |
48 | sseq1 3120 | . . . . . 6 inl inl inl inl inl inl | |
49 | foeq2 5342 | . . . . . . 7 inl inl inl inl inl inl | |
50 | 49 | exbidv 1797 | . . . . . 6 inl inl inl inl inl inl |
51 | eleq2 2203 | . . . . . . . 8 inl inl inl inl inl inl | |
52 | 51 | dcbid 823 | . . . . . . 7 inl inl inl DECID DECID inl inl inl |
53 | 52 | ralbidv 2437 | . . . . . 6 inl inl inl DECID DECID inl inl inl |
54 | 48, 50, 53 | 3anbi123d 1290 | . . . . 5 inl inl inl DECID inl inl inl inl inl inl DECID inl inl inl |
55 | 42, 54 | spcev 2780 | . . . 4 inl inl inl inl inl inl DECID inl inl inl DECID |
56 | 28, 46, 47, 55 | syl3anc 1216 | . . 3 DECID |
57 | ctssdc 6998 | . . . 4 DECID ⊔ | |
58 | foeq1 5341 | . . . . 5 ⊔ ⊔ | |
59 | 58 | cbvexv 1890 | . . . 4 ⊔ ⊔ |
60 | 57, 59 | bitri 183 | . . 3 DECID ⊔ |
61 | 56, 60 | sylib 121 | . 2 ⊔ |
62 | 5, 61 | exlimddv 1870 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 819 w3a 962 wceq 1331 wex 1468 wcel 1480 wral 2416 crab 2420 csb 3003 wss 3071 ciun 3813 class class class wbr 3929 cmpt 3989 com 4504 cxp 4537 ccnv 4538 cima 4542 ccom 4543 wfo 5121 wf1o 5122 cfv 5123 c1st 6036 c2nd 6037 c1o 6306 cen 6632 ⊔ cdju 6922 inlcinl 6930 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-xor 1354 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-1o 6313 df-er 6429 df-en 6635 df-dju 6923 df-inl 6932 df-inr 6933 df-case 6969 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-fl 10043 df-mod 10096 df-seqfrec 10219 df-exp 10293 df-dvds 11494 |
This theorem is referenced by: unct 11954 |
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