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Theorem phpm 6327
 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.)
Assertion
Ref Expression
phpm
Distinct variable groups:   ,   ,

Proof of Theorem phpm
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpr 103 . . . . . 6
2 eldifi 3066 . . . . . . . . 9
3 ne0i 3230 . . . . . . . . 9
42, 3syl 14 . . . . . . . 8
54neneqd 2226 . . . . . . 7
65ad2antlr 458 . . . . . 6
71, 6pm2.21dd 550 . . . . 5
8 php5dom 6325 . . . . . . . . . 10
98ad2antlr 458 . . . . . . . . 9
10 simplr 482 . . . . . . . . . 10
11 simpr 103 . . . . . . . . . . 11
12 vex 2560 . . . . . . . . . . . . . . . 16
1312sucex 4225 . . . . . . . . . . . . . . 15
14 difss 3070 . . . . . . . . . . . . . . 15
1513, 14ssexi 3895 . . . . . . . . . . . . . 14
16 eldifn 3067 . . . . . . . . . . . . . . . 16
1716ad3antlr 462 . . . . . . . . . . . . . . 15
18 simpllr 486 . . . . . . . . . . . . . . . . 17
1918adantr 261 . . . . . . . . . . . . . . . 16
20 simpr 103 . . . . . . . . . . . . . . . 16
2119, 20sseqtrd 2981 . . . . . . . . . . . . . . 15
22 ssdif 3078 . . . . . . . . . . . . . . . 16
23 disjsn 3432 . . . . . . . . . . . . . . . . . 18
24 disj3 3272 . . . . . . . . . . . . . . . . . 18
2523, 24bitr3i 175 . . . . . . . . . . . . . . . . 17
26 sseq1 2966 . . . . . . . . . . . . . . . . 17
2725, 26sylbi 114 . . . . . . . . . . . . . . . 16
2822, 27syl5ibr 145 . . . . . . . . . . . . . . 15
2917, 21, 28sylc 56 . . . . . . . . . . . . . 14
30 ssdomg 6258 . . . . . . . . . . . . . 14
3115, 29, 30mpsyl 59 . . . . . . . . . . . . 13
32 simplr 482 . . . . . . . . . . . . . 14
332ad3antlr 462 . . . . . . . . . . . . . . 15
3433, 20eleqtrd 2116 . . . . . . . . . . . . . 14
35 phplem3g 6319 . . . . . . . . . . . . . . 15
3635ensymd 6263 . . . . . . . . . . . . . 14
3732, 34, 36syl2anc 391 . . . . . . . . . . . . 13
38 domentr 6271 . . . . . . . . . . . . 13
3931, 37, 38syl2anc 391 . . . . . . . . . . . 12
4039adantr 261 . . . . . . . . . . 11
41 endomtr 6270 . . . . . . . . . . 11
4211, 40, 41syl2anc 391 . . . . . . . . . 10
4310, 42eqbrtrrd 3786 . . . . . . . . 9
449, 43mtand 591 . . . . . . . 8
4544ex 108 . . . . . . 7
4645rexlimdva 2433 . . . . . 6
4746imp 115 . . . . 5
48 nn0suc 4327 . . . . . 6
4948ad2antrr 457 . . . . 5
507, 47, 49mpjaodan 711 . . . 4
5150ex 108 . . 3
5251exlimdv 1700 . 2
53523impia 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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