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Mirrors > Home > ILE Home > Th. List > unennn | Unicode version |
Description: The union of two disjoint countably infinite sets is countably infinite. (Contributed by Jim Kingdon, 13-May-2022.) |
Ref | Expression |
---|---|
unennn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddennn 10984 |
. . . . . 6
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2 | 1 | ensymi 6428 |
. . . . 5
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3 | entr 6430 |
. . . . 5
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4 | 2, 3 | mpan2 416 |
. . . 4
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5 | 4 | 3ad2ant1 960 |
. . 3
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6 | evenennn 10985 |
. . . . . 6
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7 | 6 | ensymi 6428 |
. . . . 5
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8 | entr 6430 |
. . . . 5
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9 | 7, 8 | mpan2 416 |
. . . 4
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10 | 9 | 3ad2ant2 961 |
. . 3
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11 | simp3 941 |
. . 3
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12 | inrab 3254 |
. . . . 5
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13 | pm3.24 660 |
. . . . . . . 8
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14 | ancom 262 |
. . . . . . . 8
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15 | 13, 14 | mtbi 628 |
. . . . . . 7
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16 | 15 | rgenw 2424 |
. . . . . 6
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17 | rabeq0 3295 |
. . . . . 6
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18 | 16, 17 | mpbir 144 |
. . . . 5
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19 | 12, 18 | eqtri 2103 |
. . . 4
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20 | 19 | a1i 9 |
. . 3
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21 | unen 6462 |
. . 3
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22 | 5, 10, 11, 20, 21 | syl22anc 1171 |
. 2
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23 | unrab 3253 |
. . 3
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24 | rabid2 2536 |
. . . 4
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25 | nnz 8664 |
. . . . . 6
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26 | 2z 8673 |
. . . . . . 7
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27 | zdvdsdc 10596 |
. . . . . . 7
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28 | 26, 27 | mpan 415 |
. . . . . 6
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29 | exmiddc 778 |
. . . . . 6
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30 | 25, 28, 29 | 3syl 17 |
. . . . 5
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31 | 30 | orcomd 681 |
. . . 4
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32 | 24, 31 | mprgbir 2427 |
. . 3
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33 | 23, 32 | eqtr4i 2106 |
. 2
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34 | 22, 33 | syl6breq 3850 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-cnex 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-mulrcl 7346 ax-addcom 7347 ax-mulcom 7348 ax-addass 7349 ax-mulass 7350 ax-distr 7351 ax-i2m1 7352 ax-0lt1 7353 ax-1rid 7354 ax-0id 7355 ax-rnegex 7356 ax-precex 7357 ax-cnre 7358 ax-pre-ltirr 7359 ax-pre-ltwlin 7360 ax-pre-lttrn 7361 ax-pre-apti 7362 ax-pre-ltadd 7363 ax-pre-mulgt0 7364 ax-pre-mulext 7365 ax-arch 7366 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-xor 1308 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4083 df-po 4086 df-iso 4087 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-rn 4411 df-res 4412 df-ima 4413 df-iota 4933 df-fun 4970 df-fn 4971 df-f 4972 df-f1 4973 df-fo 4974 df-f1o 4975 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-1st 5845 df-2nd 5846 df-er 6221 df-en 6387 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-le 7430 df-sub 7557 df-neg 7558 df-reap 7951 df-ap 7958 df-div 8037 df-inn 8316 df-2 8374 df-n0 8565 df-z 8646 df-q 8999 df-rp 9029 df-fl 9565 df-mod 9618 df-dvds 10576 |
This theorem is referenced by: znnen 10990 |
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