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Mirrors > Home > ILE Home > Th. List > phpm | Unicode version |
Description: Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. By "proper subset" here we mean that there is an element which is in the natural number and not in the subset, or in symbols (which is stronger than not being equal in the absence of excluded middle). Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 6315 through phplem4 6318, nneneq 6320, and this final piece of the proof. (Contributed by NM, 29-May-1998.) |
Ref | Expression |
---|---|
phpm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 103 | . . . . . 6 | |
2 | eldifi 3066 | . . . . . . . . 9 | |
3 | ne0i 3230 | . . . . . . . . 9 | |
4 | 2, 3 | syl 14 | . . . . . . . 8 |
5 | 4 | neneqd 2226 | . . . . . . 7 |
6 | 5 | ad2antlr 458 | . . . . . 6 |
7 | 1, 6 | pm2.21dd 550 | . . . . 5 |
8 | php5dom 6325 | . . . . . . . . . 10 | |
9 | 8 | ad2antlr 458 | . . . . . . . . 9 |
10 | simplr 482 | . . . . . . . . . 10 | |
11 | simpr 103 | . . . . . . . . . . 11 | |
12 | vex 2560 | . . . . . . . . . . . . . . . 16 | |
13 | 12 | sucex 4225 | . . . . . . . . . . . . . . 15 |
14 | difss 3070 | . . . . . . . . . . . . . . 15 | |
15 | 13, 14 | ssexi 3895 | . . . . . . . . . . . . . 14 |
16 | eldifn 3067 | . . . . . . . . . . . . . . . 16 | |
17 | 16 | ad3antlr 462 | . . . . . . . . . . . . . . 15 |
18 | simpllr 486 | . . . . . . . . . . . . . . . . 17 | |
19 | 18 | adantr 261 | . . . . . . . . . . . . . . . 16 |
20 | simpr 103 | . . . . . . . . . . . . . . . 16 | |
21 | 19, 20 | sseqtrd 2981 | . . . . . . . . . . . . . . 15 |
22 | ssdif 3078 | . . . . . . . . . . . . . . . 16 | |
23 | disjsn 3432 | . . . . . . . . . . . . . . . . . 18 | |
24 | disj3 3272 | . . . . . . . . . . . . . . . . . 18 | |
25 | 23, 24 | bitr3i 175 | . . . . . . . . . . . . . . . . 17 |
26 | sseq1 2966 | . . . . . . . . . . . . . . . . 17 | |
27 | 25, 26 | sylbi 114 | . . . . . . . . . . . . . . . 16 |
28 | 22, 27 | syl5ibr 145 | . . . . . . . . . . . . . . 15 |
29 | 17, 21, 28 | sylc 56 | . . . . . . . . . . . . . 14 |
30 | ssdomg 6258 | . . . . . . . . . . . . . 14 | |
31 | 15, 29, 30 | mpsyl 59 | . . . . . . . . . . . . 13 |
32 | simplr 482 | . . . . . . . . . . . . . 14 | |
33 | 2 | ad3antlr 462 | . . . . . . . . . . . . . . 15 |
34 | 33, 20 | eleqtrd 2116 | . . . . . . . . . . . . . 14 |
35 | phplem3g 6319 | . . . . . . . . . . . . . . 15 | |
36 | 35 | ensymd 6263 | . . . . . . . . . . . . . 14 |
37 | 32, 34, 36 | syl2anc 391 | . . . . . . . . . . . . 13 |
38 | domentr 6271 | . . . . . . . . . . . . 13 | |
39 | 31, 37, 38 | syl2anc 391 | . . . . . . . . . . . 12 |
40 | 39 | adantr 261 | . . . . . . . . . . 11 |
41 | endomtr 6270 | . . . . . . . . . . 11 | |
42 | 11, 40, 41 | syl2anc 391 | . . . . . . . . . 10 |
43 | 10, 42 | eqbrtrrd 3786 | . . . . . . . . 9 |
44 | 9, 43 | mtand 591 | . . . . . . . 8 |
45 | 44 | ex 108 | . . . . . . 7 |
46 | 45 | rexlimdva 2433 | . . . . . 6 |
47 | 46 | imp 115 | . . . . 5 |
48 | nn0suc 4327 | . . . . . 6 | |
49 | 48 | ad2antrr 457 | . . . . 5 |
50 | 7, 47, 49 | mpjaodan 711 | . . . 4 |
51 | 50 | ex 108 | . . 3 |
52 | 51 | exlimdv 1700 | . 2 |
53 | 52 | 3impia 1101 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 w3a 885 wceq 1243 wex 1381 wcel 1393 wne 2204 wrex 2307 cvv 2557 cdif 2914 cin 2916 wss 2917 c0 3224 csn 3375 class class class wbr 3764 csuc 4102 com 4313 cen 6219 cdom 6220 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-er 6106 df-en 6222 df-dom 6223 |
This theorem is referenced by: phpelm 6328 |
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