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Mirrors > Home > ILE Home > Th. List > qnnen | Unicode version |
Description: The rational numbers are countably infinite. Corollary 8.1.23 of [AczelRathjen], p. 75. This is Metamath 100 proof #3. (Contributed by Jim Kingdon, 11-Aug-2023.) |
Ref | Expression |
---|---|
qnnen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qdceq 10203 | . . 3 DECID | |
2 | 1 | rgen2a 2524 | . 2 DECID |
3 | znnen 12353 | . . . . . . . 8 | |
4 | nnex 8884 | . . . . . . . . 9 | |
5 | 4 | enref 6743 | . . . . . . . 8 |
6 | xpen 6823 | . . . . . . . 8 | |
7 | 3, 5, 6 | mp2an 424 | . . . . . . 7 |
8 | xpnnen 12349 | . . . . . . 7 | |
9 | 7, 8 | entri 6764 | . . . . . 6 |
10 | nnenom 10390 | . . . . . 6 | |
11 | 9, 10 | entri 6764 | . . . . 5 |
12 | 11 | ensymi 6760 | . . . 4 |
13 | bren 6725 | . . . 4 | |
14 | 12, 13 | mpbi 144 | . . 3 |
15 | f1ofo 5449 | . . . . 5 | |
16 | divfnzn 9580 | . . . . . . . . 9 | |
17 | fnfun 5295 | . . . . . . . . 9 | |
18 | 16, 17 | ax-mp 5 | . . . . . . . 8 |
19 | fndm 5297 | . . . . . . . . 9 | |
20 | eqimss2 3202 | . . . . . . . . 9 | |
21 | 16, 19, 20 | mp2b 8 | . . . . . . . 8 |
22 | fores 5429 | . . . . . . . 8 | |
23 | 18, 21, 22 | mp2an 424 | . . . . . . 7 |
24 | resima 4924 | . . . . . . . . 9 | |
25 | df-q 9579 | . . . . . . . . 9 | |
26 | 24, 25 | eqtr4i 2194 | . . . . . . . 8 |
27 | foeq3 5418 | . . . . . . . 8 | |
28 | 26, 27 | ax-mp 5 | . . . . . . 7 |
29 | 23, 28 | mpbi 144 | . . . . . 6 |
30 | foco 5430 | . . . . . 6 | |
31 | 29, 30 | mpan 422 | . . . . 5 |
32 | zex 9221 | . . . . . . . . 9 | |
33 | 32, 4 | xpex 4726 | . . . . . . . 8 |
34 | resfunexg 5717 | . . . . . . . 8 | |
35 | 18, 33, 34 | mp2an 424 | . . . . . . 7 |
36 | vex 2733 | . . . . . . 7 | |
37 | 35, 36 | coex 5156 | . . . . . 6 |
38 | foeq1 5416 | . . . . . 6 | |
39 | 37, 38 | spcev 2825 | . . . . 5 |
40 | 15, 31, 39 | 3syl 17 | . . . 4 |
41 | 40 | exlimiv 1591 | . . 3 |
42 | 14, 41 | ax-mp 5 | . 2 |
43 | 10 | ensymi 6760 | . . 3 |
44 | qex 9591 | . . . 4 | |
45 | nnssq 9588 | . . . 4 | |
46 | ssdomg 6756 | . . . 4 | |
47 | 44, 45, 46 | mp2 16 | . . 3 |
48 | endomtr 6768 | . . 3 | |
49 | 43, 47, 48 | mp2an 424 | . 2 |
50 | ctinf 12385 | . 2 DECID | |
51 | 2, 42, 49, 50 | mpbir3an 1174 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 104 DECID wdc 829 wceq 1348 wex 1485 wcel 2141 wral 2448 cvv 2730 wss 3121 class class class wbr 3989 com 4574 cxp 4609 cdm 4611 cres 4613 cima 4614 ccom 4615 wfun 5192 wfn 5193 wfo 5196 wf1o 5197 cen 6716 cdom 6717 cdiv 8589 cn 8878 cz 9212 cq 9578 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-xor 1371 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-1o 6395 df-er 6513 df-pm 6629 df-en 6719 df-dom 6720 df-fin 6721 df-dju 7015 df-inl 7024 df-inr 7025 df-case 7061 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-dvds 11750 |
This theorem is referenced by: (None) |
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