Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6746 through phplem4 6749, nneneq 6751, and this final piece of the proof. (Contributed by NM, 29-May-1998.) |
Ref | Expression |
---|---|
phpm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . . 6 | |
2 | eldifi 3198 | . . . . . . . . 9 | |
3 | ne0i 3369 | . . . . . . . . 9 | |
4 | 2, 3 | syl 14 | . . . . . . . 8 |
5 | 4 | neneqd 2329 | . . . . . . 7 |
6 | 5 | ad2antlr 480 | . . . . . 6 |
7 | 1, 6 | pm2.21dd 609 | . . . . 5 |
8 | php5dom 6757 | . . . . . . . . . 10 | |
9 | 8 | ad2antlr 480 | . . . . . . . . 9 |
10 | simplr 519 | . . . . . . . . . 10 | |
11 | simpr 109 | . . . . . . . . . . 11 | |
12 | vex 2689 | . . . . . . . . . . . . . . . 16 | |
13 | 12 | sucex 4415 | . . . . . . . . . . . . . . 15 |
14 | difss 3202 | . . . . . . . . . . . . . . 15 | |
15 | 13, 14 | ssexi 4066 | . . . . . . . . . . . . . 14 |
16 | eldifn 3199 | . . . . . . . . . . . . . . . 16 | |
17 | 16 | ad3antlr 484 | . . . . . . . . . . . . . . 15 |
18 | simpllr 523 | . . . . . . . . . . . . . . . . 17 | |
19 | 18 | adantr 274 | . . . . . . . . . . . . . . . 16 |
20 | simpr 109 | . . . . . . . . . . . . . . . 16 | |
21 | 19, 20 | sseqtrd 3135 | . . . . . . . . . . . . . . 15 |
22 | ssdif 3211 | . . . . . . . . . . . . . . . 16 | |
23 | disjsn 3585 | . . . . . . . . . . . . . . . . . 18 | |
24 | disj3 3415 | . . . . . . . . . . . . . . . . . 18 | |
25 | 23, 24 | bitr3i 185 | . . . . . . . . . . . . . . . . 17 |
26 | sseq1 3120 | . . . . . . . . . . . . . . . . 17 | |
27 | 25, 26 | sylbi 120 | . . . . . . . . . . . . . . . 16 |
28 | 22, 27 | syl5ibr 155 | . . . . . . . . . . . . . . 15 |
29 | 17, 21, 28 | sylc 62 | . . . . . . . . . . . . . 14 |
30 | ssdomg 6672 | . . . . . . . . . . . . . 14 | |
31 | 15, 29, 30 | mpsyl 65 | . . . . . . . . . . . . 13 |
32 | simplr 519 | . . . . . . . . . . . . . 14 | |
33 | 2 | ad3antlr 484 | . . . . . . . . . . . . . . 15 |
34 | 33, 20 | eleqtrd 2218 | . . . . . . . . . . . . . 14 |
35 | phplem3g 6750 | . . . . . . . . . . . . . . 15 | |
36 | 35 | ensymd 6677 | . . . . . . . . . . . . . 14 |
37 | 32, 34, 36 | syl2anc 408 | . . . . . . . . . . . . 13 |
38 | domentr 6685 | . . . . . . . . . . . . 13 | |
39 | 31, 37, 38 | syl2anc 408 | . . . . . . . . . . . 12 |
40 | 39 | adantr 274 | . . . . . . . . . . 11 |
41 | endomtr 6684 | . . . . . . . . . . 11 | |
42 | 11, 40, 41 | syl2anc 408 | . . . . . . . . . 10 |
43 | 10, 42 | eqbrtrrd 3952 | . . . . . . . . 9 |
44 | 9, 43 | mtand 654 | . . . . . . . 8 |
45 | 44 | ex 114 | . . . . . . 7 |
46 | 45 | rexlimdva 2549 | . . . . . 6 |
47 | 46 | imp 123 | . . . . 5 |
48 | nn0suc 4518 | . . . . . 6 | |
49 | 48 | ad2antrr 479 | . . . . 5 |
50 | 7, 47, 49 | mpjaodan 787 | . . . 4 |
51 | 50 | ex 114 | . . 3 |
52 | 51 | exlimdv 1791 | . 2 |
53 | 52 | 3impia 1178 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 w3a 962 wceq 1331 wex 1468 wcel 1480 wne 2308 wrex 2417 cvv 2686 cdif 3068 cin 3070 wss 3071 c0 3363 csn 3527 class class class wbr 3929 csuc 4287 com 4504 cen 6632 cdom 6633 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-er 6429 df-en 6635 df-dom 6636 |
This theorem is referenced by: phpelm 6760 |
Copyright terms: Public domain | W3C validator |