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Mirrors > Home > ILE Home > Th. List > unct | Unicode version |
Description: The union of two countable sets is countable. (Contributed by Jim Kingdon, 1-Nov-2023.) |
Ref | Expression |
---|---|
unct | ⊔ ⊔ ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2onn 6417 | . . . . . . . 8 | |
2 | nnfi 6766 | . . . . . . . 8 | |
3 | finct 7001 | . . . . . . . 8 ⊔ | |
4 | 1, 2, 3 | mp2b 8 | . . . . . . 7 ⊔ |
5 | 4 | a1i 9 | . . . . . 6 ⊔ ⊔ ⊔ |
6 | simpr 109 | . . . . . . . . 9 ⊔ ⊔ ⊔ ⊔ | |
7 | df2o3 6327 | . . . . . . . . . 10 | |
8 | djueq1 6925 | . . . . . . . . . 10 ⊔ ⊔ | |
9 | foeq3 5343 | . . . . . . . . . 10 ⊔ ⊔ ⊔ ⊔ | |
10 | 7, 8, 9 | mp2b 8 | . . . . . . . . 9 ⊔ ⊔ |
11 | 6, 10 | sylib 121 | . . . . . . . 8 ⊔ ⊔ ⊔ ⊔ |
12 | simplll 522 | . . . . . . . . . . . 12 ⊔ ⊔ ⊔ ⊔ | |
13 | iftrue 3479 | . . . . . . . . . . . . . 14 | |
14 | eqidd 2140 | . . . . . . . . . . . . . 14 | |
15 | iftrue 3479 | . . . . . . . . . . . . . . 15 | |
16 | djueq1 6925 | . . . . . . . . . . . . . . 15 ⊔ ⊔ | |
17 | 15, 16 | syl 14 | . . . . . . . . . . . . . 14 ⊔ ⊔ |
18 | 13, 14, 17 | foeq123d 5361 | . . . . . . . . . . . . 13 ⊔ ⊔ |
19 | 18 | adantl 275 | . . . . . . . . . . . 12 ⊔ ⊔ ⊔ ⊔ ⊔ |
20 | 12, 19 | mpbird 166 | . . . . . . . . . . 11 ⊔ ⊔ ⊔ ⊔ |
21 | 20 | ex 114 | . . . . . . . . . 10 ⊔ ⊔ ⊔ ⊔ |
22 | simpllr 523 | . . . . . . . . . . . 12 ⊔ ⊔ ⊔ ⊔ | |
23 | 1n0 6329 | . . . . . . . . . . . . . . . 16 | |
24 | 23 | neii 2310 | . . . . . . . . . . . . . . 15 |
25 | eqeq1 2146 | . . . . . . . . . . . . . . 15 | |
26 | 24, 25 | mtbiri 664 | . . . . . . . . . . . . . 14 |
27 | 26 | adantl 275 | . . . . . . . . . . . . 13 ⊔ ⊔ ⊔ |
28 | iffalse 3482 | . . . . . . . . . . . . . 14 | |
29 | eqidd 2140 | . . . . . . . . . . . . . 14 | |
30 | iffalse 3482 | . . . . . . . . . . . . . . 15 | |
31 | djueq1 6925 | . . . . . . . . . . . . . . 15 ⊔ ⊔ | |
32 | 30, 31 | syl 14 | . . . . . . . . . . . . . 14 ⊔ ⊔ |
33 | 28, 29, 32 | foeq123d 5361 | . . . . . . . . . . . . 13 ⊔ ⊔ |
34 | 27, 33 | syl 14 | . . . . . . . . . . . 12 ⊔ ⊔ ⊔ ⊔ ⊔ |
35 | 22, 34 | mpbird 166 | . . . . . . . . . . 11 ⊔ ⊔ ⊔ ⊔ |
36 | 35 | ex 114 | . . . . . . . . . 10 ⊔ ⊔ ⊔ ⊔ |
37 | 21, 36 | jaod 706 | . . . . . . . . 9 ⊔ ⊔ ⊔ ⊔ |
38 | elpri 3550 | . . . . . . . . 9 | |
39 | 37, 38 | impel 278 | . . . . . . . 8 ⊔ ⊔ ⊔ ⊔ |
40 | 11, 39 | ctiunct 11953 | . . . . . . 7 ⊔ ⊔ ⊔ ⊔ |
41 | 0lt2o 6338 | . . . . . . . . . 10 | |
42 | 1lt2o 6339 | . . . . . . . . . 10 | |
43 | 26 | iffalsed 3484 | . . . . . . . . . . 11 |
44 | 15, 43 | iunxprg 3893 | . . . . . . . . . 10 |
45 | 41, 42, 44 | mp2an 422 | . . . . . . . . 9 |
46 | djueq1 6925 | . . . . . . . . 9 ⊔ ⊔ | |
47 | foeq3 5343 | . . . . . . . . 9 ⊔ ⊔ ⊔ ⊔ | |
48 | 45, 46, 47 | mp2b 8 | . . . . . . . 8 ⊔ ⊔ |
49 | 48 | exbii 1584 | . . . . . . 7 ⊔ ⊔ |
50 | 40, 49 | sylib 121 | . . . . . 6 ⊔ ⊔ ⊔ ⊔ |
51 | 5, 50 | exlimddv 1870 | . . . . 5 ⊔ ⊔ ⊔ |
52 | 51 | ex 114 | . . . 4 ⊔ ⊔ ⊔ |
53 | 52 | exlimiv 1577 | . . 3 ⊔ ⊔ ⊔ |
54 | 53 | exlimdv 1791 | . 2 ⊔ ⊔ ⊔ |
55 | 54 | imp 123 | 1 ⊔ ⊔ ⊔ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wex 1468 wcel 1480 cun 3069 c0 3363 cif 3474 cpr 3528 ciun 3813 com 4504 wfo 5121 c1o 6306 c2o 6307 cfn 6634 ⊔ cdju 6922 |
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-2o 6314 df-er 6429 df-en 6635 df-fin 6637 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: (None) |
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