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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 6338 through phplem4 6341, nneneq 6343, and this final piece of the proof. (Contributed by NM, 29-May-1998.) |
Ref | Expression |
---|---|
phpm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 108 | . . . . . 6 | |
2 | eldifi 3094 | . . . . . . . . 9 | |
3 | ne0i 3257 | . . . . . . . . 9 | |
4 | 2, 3 | syl 14 | . . . . . . . 8 |
5 | 4 | neneqd 2266 | . . . . . . 7 |
6 | 5 | ad2antlr 472 | . . . . . 6 |
7 | 1, 6 | pm2.21dd 582 | . . . . 5 |
8 | php5dom 6349 | . . . . . . . . . 10 | |
9 | 8 | ad2antlr 472 | . . . . . . . . 9 |
10 | simplr 496 | . . . . . . . . . 10 | |
11 | simpr 108 | . . . . . . . . . . 11 | |
12 | vex 2604 | . . . . . . . . . . . . . . . 16 | |
13 | 12 | sucex 4243 | . . . . . . . . . . . . . . 15 |
14 | difss 3098 | . . . . . . . . . . . . . . 15 | |
15 | 13, 14 | ssexi 3916 | . . . . . . . . . . . . . 14 |
16 | eldifn 3095 | . . . . . . . . . . . . . . . 16 | |
17 | 16 | ad3antlr 476 | . . . . . . . . . . . . . . 15 |
18 | simpllr 500 | . . . . . . . . . . . . . . . . 17 | |
19 | 18 | adantr 270 | . . . . . . . . . . . . . . . 16 |
20 | simpr 108 | . . . . . . . . . . . . . . . 16 | |
21 | 19, 20 | sseqtrd 3035 | . . . . . . . . . . . . . . 15 |
22 | ssdif 3107 | . . . . . . . . . . . . . . . 16 | |
23 | disjsn 3454 | . . . . . . . . . . . . . . . . . 18 | |
24 | disj3 3296 | . . . . . . . . . . . . . . . . . 18 | |
25 | 23, 24 | bitr3i 184 | . . . . . . . . . . . . . . . . 17 |
26 | sseq1 3020 | . . . . . . . . . . . . . . . . 17 | |
27 | 25, 26 | sylbi 119 | . . . . . . . . . . . . . . . 16 |
28 | 22, 27 | syl5ibr 154 | . . . . . . . . . . . . . . 15 |
29 | 17, 21, 28 | sylc 61 | . . . . . . . . . . . . . 14 |
30 | ssdomg 6281 | . . . . . . . . . . . . . 14 | |
31 | 15, 29, 30 | mpsyl 64 | . . . . . . . . . . . . 13 |
32 | simplr 496 | . . . . . . . . . . . . . 14 | |
33 | 2 | ad3antlr 476 | . . . . . . . . . . . . . . 15 |
34 | 33, 20 | eleqtrd 2157 | . . . . . . . . . . . . . 14 |
35 | phplem3g 6342 | . . . . . . . . . . . . . . 15 | |
36 | 35 | ensymd 6286 | . . . . . . . . . . . . . 14 |
37 | 32, 34, 36 | syl2anc 403 | . . . . . . . . . . . . 13 |
38 | domentr 6294 | . . . . . . . . . . . . 13 | |
39 | 31, 37, 38 | syl2anc 403 | . . . . . . . . . . . 12 |
40 | 39 | adantr 270 | . . . . . . . . . . 11 |
41 | endomtr 6293 | . . . . . . . . . . 11 | |
42 | 11, 40, 41 | syl2anc 403 | . . . . . . . . . 10 |
43 | 10, 42 | eqbrtrrd 3807 | . . . . . . . . 9 |
44 | 9, 43 | mtand 623 | . . . . . . . 8 |
45 | 44 | ex 113 | . . . . . . 7 |
46 | 45 | rexlimdva 2477 | . . . . . 6 |
47 | 46 | imp 122 | . . . . 5 |
48 | nn0suc 4345 | . . . . . 6 | |
49 | 48 | ad2antrr 471 | . . . . 5 |
50 | 7, 47, 49 | mpjaodan 744 | . . . 4 |
51 | 50 | ex 113 | . . 3 |
52 | 51 | exlimdv 1740 | . 2 |
53 | 52 | 3impia 1135 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 w3a 919 wceq 1284 wex 1421 wcel 1433 wne 2245 wrex 2349 cvv 2601 cdif 2970 cin 2972 wss 2973 c0 3251 csn 3398 class class class wbr 3785 csuc 4120 com 4331 cen 6242 cdom 6243 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-er 6129 df-en 6245 df-dom 6246 |
This theorem is referenced by: phpelm 6352 |
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