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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 10141 | . . 3 DECID | |
2 | 1 | rgen2a 2511 | . 2 DECID |
3 | znnen 12114 | . . . . . . . 8 | |
4 | nnex 8834 | . . . . . . . . 9 | |
5 | 4 | enref 6707 | . . . . . . . 8 |
6 | xpen 6787 | . . . . . . . 8 | |
7 | 3, 5, 6 | mp2an 423 | . . . . . . 7 |
8 | xpnnen 12110 | . . . . . . 7 | |
9 | 7, 8 | entri 6728 | . . . . . 6 |
10 | nnenom 10328 | . . . . . 6 | |
11 | 9, 10 | entri 6728 | . . . . 5 |
12 | 11 | ensymi 6724 | . . . 4 |
13 | bren 6689 | . . . 4 | |
14 | 12, 13 | mpbi 144 | . . 3 |
15 | f1ofo 5420 | . . . . 5 | |
16 | divfnzn 9525 | . . . . . . . . 9 | |
17 | fnfun 5266 | . . . . . . . . 9 | |
18 | 16, 17 | ax-mp 5 | . . . . . . . 8 |
19 | fndm 5268 | . . . . . . . . 9 | |
20 | eqimss2 3183 | . . . . . . . . 9 | |
21 | 16, 19, 20 | mp2b 8 | . . . . . . . 8 |
22 | fores 5400 | . . . . . . . 8 | |
23 | 18, 21, 22 | mp2an 423 | . . . . . . 7 |
24 | resima 4898 | . . . . . . . . 9 | |
25 | df-q 9524 | . . . . . . . . 9 | |
26 | 24, 25 | eqtr4i 2181 | . . . . . . . 8 |
27 | foeq3 5389 | . . . . . . . 8 | |
28 | 26, 27 | ax-mp 5 | . . . . . . 7 |
29 | 23, 28 | mpbi 144 | . . . . . 6 |
30 | foco 5401 | . . . . . 6 | |
31 | 29, 30 | mpan 421 | . . . . 5 |
32 | zex 9171 | . . . . . . . . 9 | |
33 | 32, 4 | xpex 4700 | . . . . . . . 8 |
34 | resfunexg 5687 | . . . . . . . 8 | |
35 | 18, 33, 34 | mp2an 423 | . . . . . . 7 |
36 | vex 2715 | . . . . . . 7 | |
37 | 35, 36 | coex 5130 | . . . . . 6 |
38 | foeq1 5387 | . . . . . 6 | |
39 | 37, 38 | spcev 2807 | . . . . 5 |
40 | 15, 31, 39 | 3syl 17 | . . . 4 |
41 | 40 | exlimiv 1578 | . . 3 |
42 | 14, 41 | ax-mp 5 | . 2 |
43 | 10 | ensymi 6724 | . . 3 |
44 | qex 9536 | . . . 4 | |
45 | nnssq 9533 | . . . 4 | |
46 | ssdomg 6720 | . . . 4 | |
47 | 44, 45, 46 | mp2 16 | . . 3 |
48 | endomtr 6732 | . . 3 | |
49 | 43, 47, 48 | mp2an 423 | . 2 |
50 | ctinf 12146 | . 2 DECID | |
51 | 2, 42, 49, 50 | mpbir3an 1164 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 104 DECID wdc 820 wceq 1335 wex 1472 wcel 2128 wral 2435 cvv 2712 wss 3102 class class class wbr 3965 com 4548 cxp 4583 cdm 4585 cres 4587 cima 4588 ccom 4589 wfun 5163 wfn 5164 wfo 5167 wf1o 5168 cen 6680 cdom 6681 cdiv 8540 cn 8828 cz 9162 cq 9523 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-mulrcl 7826 ax-addcom 7827 ax-mulcom 7828 ax-addass 7829 ax-mulass 7830 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-1rid 7834 ax-0id 7835 ax-rnegex 7836 ax-precex 7837 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-apti 7842 ax-pre-ltadd 7843 ax-pre-mulgt0 7844 ax-pre-mulext 7845 ax-arch 7846 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-xor 1358 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-frec 6335 df-1o 6360 df-er 6477 df-pm 6593 df-en 6683 df-dom 6684 df-fin 6685 df-dju 6977 df-inl 6986 df-inr 6987 df-case 7023 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-reap 8445 df-ap 8452 df-div 8541 df-inn 8829 df-2 8887 df-n0 9086 df-z 9163 df-uz 9435 df-q 9524 df-rp 9556 df-fz 9908 df-fl 10164 df-mod 10217 df-seqfrec 10340 df-exp 10414 df-dvds 11679 |
This theorem is referenced by: (None) |
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