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Theorem php 7061
Description: Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. 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 7056 through phplem4 7059, nneneq 7060, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
Assertion
Ref Expression
php  |-  ( ( A  e.  om  /\  B  C.  A )  ->  -.  A  ~~  B )

Proof of Theorem php
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 3496 . . . . . . . 8  |-  (/)  C_  B
2 sspsstr 3294 . . . . . . . 8  |-  ( (
(/)  C_  B  /\  B  C.  A )  ->  (/)  C.  A
)
31, 2mpan 651 . . . . . . 7  |-  ( B 
C.  A  ->  (/)  C.  A
)
4 0pss 3505 . . . . . . . 8  |-  ( (/)  C.  A  <->  A  =/=  (/) )
5 df-ne 2461 . . . . . . . 8  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
64, 5bitri 240 . . . . . . 7  |-  ( (/)  C.  A  <->  -.  A  =  (/) )
73, 6sylib 188 . . . . . 6  |-  ( B 
C.  A  ->  -.  A  =  (/) )
8 nn0suc 4696 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
98orcanai 879 . . . . . 6  |-  ( ( A  e.  om  /\  -.  A  =  (/) )  ->  E. x  e.  om  A  =  suc  x )
107, 9sylan2 460 . . . . 5  |-  ( ( A  e.  om  /\  B  C.  A )  ->  E. x  e.  om  A  =  suc  x )
11 pssnel 3532 . . . . . . . . . 10  |-  ( B 
C.  suc  x  ->  E. y ( y  e. 
suc  x  /\  -.  y  e.  B )
)
12 pssss 3284 . . . . . . . . . . . . . . . . 17  |-  ( B 
C.  suc  x  ->  B 
C_  suc  x )
13 ssdif 3324 . . . . . . . . . . . . . . . . . 18  |-  ( B 
C_  suc  x  ->  ( B  \  { y } )  C_  ( suc  x  \  { y } ) )
14 disjsn 3706 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  i^i  { y } )  =  (/)  <->  -.  y  e.  B )
15 disj3 3512 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  i^i  { y } )  =  (/)  <->  B  =  ( B  \  { y } ) )
1614, 15bitr3i 242 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  y  e.  B  <->  B  =  ( B  \  { y } ) )
17 sseq1 3212 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =  ( B  \  { y } )  ->  ( B  C_  ( suc  x  \  {
y } )  <->  ( B  \  { y } ) 
C_  ( suc  x  \  { y } ) ) )
1816, 17sylbi 187 . . . . . . . . . . . . . . . . . 18  |-  ( -.  y  e.  B  -> 
( B  C_  ( suc  x  \  { y } )  <->  ( B  \  { y } ) 
C_  ( suc  x  \  { y } ) ) )
1913, 18syl5ibr 212 . . . . . . . . . . . . . . . . 17  |-  ( -.  y  e.  B  -> 
( B  C_  suc  x  ->  B  C_  ( suc  x  \  { y } ) ) )
20 vex 2804 . . . . . . . . . . . . . . . . . . . 20  |-  x  e. 
_V
2120sucex 4618 . . . . . . . . . . . . . . . . . . 19  |-  suc  x  e.  _V
22 difss 3316 . . . . . . . . . . . . . . . . . . 19  |-  ( suc  x  \  { y } )  C_  suc  x
2321, 22ssexi 4175 . . . . . . . . . . . . . . . . . 18  |-  ( suc  x  \  { y } )  e.  _V
24 ssdomg 6923 . . . . . . . . . . . . . . . . . 18  |-  ( ( suc  x  \  {
y } )  e. 
_V  ->  ( B  C_  ( suc  x  \  {
y } )  ->  B  ~<_  ( suc  x  \  { y } ) ) )
2523, 24ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  ( B 
C_  ( suc  x  \  { y } )  ->  B  ~<_  ( suc  x  \  { y } ) )
2612, 19, 25syl56 30 . . . . . . . . . . . . . . . 16  |-  ( -.  y  e.  B  -> 
( B  C.  suc  x  ->  B  ~<_  ( suc  x  \  { y } ) ) )
2726imp 418 . . . . . . . . . . . . . . 15  |-  ( ( -.  y  e.  B  /\  B  C.  suc  x
)  ->  B  ~<_  ( suc  x  \  { y } ) )
28 vex 2804 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
2920, 28phplem3 7058 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  om  /\  y  e.  suc  x )  ->  x  ~~  ( suc  x  \  { y } ) )
30 ensym 6926 . . . . . . . . . . . . . . . 16  |-  ( x 
~~  ( suc  x  \  { y } )  ->  ( suc  x  \  { y } ) 
~~  x )
3129, 30syl 15 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  om  /\  y  e.  suc  x )  ->  ( suc  x  \  { y } ) 
~~  x )
32 domentr 6936 . . . . . . . . . . . . . . 15  |-  ( ( B  ~<_  ( suc  x  \  { y } )  /\  ( suc  x  \  { y } ) 
~~  x )  ->  B  ~<_  x )
3327, 31, 32syl2an 463 . . . . . . . . . . . . . 14  |-  ( ( ( -.  y  e.  B  /\  B  C.  suc  x )  /\  (
x  e.  om  /\  y  e.  suc  x ) )  ->  B  ~<_  x )
3433exp43 595 . . . . . . . . . . . . 13  |-  ( -.  y  e.  B  -> 
( B  C.  suc  x  ->  ( x  e. 
om  ->  ( y  e. 
suc  x  ->  B  ~<_  x ) ) ) )
3534com4r 80 . . . . . . . . . . . 12  |-  ( y  e.  suc  x  -> 
( -.  y  e.  B  ->  ( B  C.  suc  x  ->  (
x  e.  om  ->  B  ~<_  x ) ) ) )
3635imp 418 . . . . . . . . . . 11  |-  ( ( y  e.  suc  x  /\  -.  y  e.  B
)  ->  ( B  C.  suc  x  ->  (
x  e.  om  ->  B  ~<_  x ) ) )
3736exlimiv 1624 . . . . . . . . . 10  |-  ( E. y ( y  e. 
suc  x  /\  -.  y  e.  B )  ->  ( B  C.  suc  x  ->  ( x  e. 
om  ->  B  ~<_  x ) ) )
3811, 37mpcom 32 . . . . . . . . 9  |-  ( B 
C.  suc  x  ->  ( x  e.  om  ->  B  ~<_  x ) )
39 endomtr 6935 . . . . . . . . . . . 12  |-  ( ( suc  x  ~~  B  /\  B  ~<_  x )  ->  suc  x  ~<_  x )
40 sssucid 4485 . . . . . . . . . . . . 13  |-  x  C_  suc  x
41 ssdomg 6923 . . . . . . . . . . . . 13  |-  ( suc  x  e.  _V  ->  ( x  C_  suc  x  ->  x  ~<_  suc  x )
)
4221, 40, 41mp2 17 . . . . . . . . . . . 12  |-  x  ~<_  suc  x
43 sbth 6997 . . . . . . . . . . . 12  |-  ( ( suc  x  ~<_  x  /\  x  ~<_  suc  x )  ->  suc  x  ~~  x
)
4439, 42, 43sylancl 643 . . . . . . . . . . 11  |-  ( ( suc  x  ~~  B  /\  B  ~<_  x )  ->  suc  x  ~~  x
)
4544expcom 424 . . . . . . . . . 10  |-  ( B  ~<_  x  ->  ( suc  x  ~~  B  ->  suc  x  ~~  x ) )
46 peano2b 4688 . . . . . . . . . . . . 13  |-  ( x  e.  om  <->  suc  x  e. 
om )
47 nnord 4680 . . . . . . . . . . . . 13  |-  ( suc  x  e.  om  ->  Ord 
suc  x )
4846, 47sylbi 187 . . . . . . . . . . . 12  |-  ( x  e.  om  ->  Ord  suc  x )
4920sucid 4487 . . . . . . . . . . . 12  |-  x  e. 
suc  x
50 nordeq 4427 . . . . . . . . . . . 12  |-  ( ( Ord  suc  x  /\  x  e.  suc  x )  ->  suc  x  =/=  x )
5148, 49, 50sylancl 643 . . . . . . . . . . 11  |-  ( x  e.  om  ->  suc  x  =/=  x )
52 nneneq 7060 . . . . . . . . . . . . . 14  |-  ( ( suc  x  e.  om  /\  x  e.  om )  ->  ( suc  x  ~~  x 
<->  suc  x  =  x ) )
5346, 52sylanb 458 . . . . . . . . . . . . 13  |-  ( ( x  e.  om  /\  x  e.  om )  ->  ( suc  x  ~~  x 
<->  suc  x  =  x ) )
5453anidms 626 . . . . . . . . . . . 12  |-  ( x  e.  om  ->  ( suc  x  ~~  x  <->  suc  x  =  x ) )
5554necon3bbid 2493 . . . . . . . . . . 11  |-  ( x  e.  om  ->  ( -.  suc  x  ~~  x  <->  suc  x  =/=  x ) )
5651, 55mpbird 223 . . . . . . . . . 10  |-  ( x  e.  om  ->  -.  suc  x  ~~  x )
5745, 56nsyli 133 . . . . . . . . 9  |-  ( B  ~<_  x  ->  ( x  e.  om  ->  -.  suc  x  ~~  B ) )
5838, 57syli 33 . . . . . . . 8  |-  ( B 
C.  suc  x  ->  ( x  e.  om  ->  -. 
suc  x  ~~  B
) )
5958com12 27 . . . . . . 7  |-  ( x  e.  om  ->  ( B  C.  suc  x  ->  -.  suc  x  ~~  B
) )
60 psseq2 3277 . . . . . . . 8  |-  ( A  =  suc  x  -> 
( B  C.  A  <->  B 
C.  suc  x )
)
61 breq1 4042 . . . . . . . . 9  |-  ( A  =  suc  x  -> 
( A  ~~  B  <->  suc  x  ~~  B ) )
6261notbid 285 . . . . . . . 8  |-  ( A  =  suc  x  -> 
( -.  A  ~~  B 
<->  -.  suc  x  ~~  B ) )
6360, 62imbi12d 311 . . . . . . 7  |-  ( A  =  suc  x  -> 
( ( B  C.  A  ->  -.  A  ~~  B )  <->  ( B  C.  suc  x  ->  -.  suc  x  ~~  B ) ) )
6459, 63syl5ibrcom 213 . . . . . 6  |-  ( x  e.  om  ->  ( A  =  suc  x  -> 
( B  C.  A  ->  -.  A  ~~  B
) ) )
6564rexlimiv 2674 . . . . 5  |-  ( E. x  e.  om  A  =  suc  x  ->  ( B  C.  A  ->  -.  A  ~~  B ) )
6610, 65syl 15 . . . 4  |-  ( ( A  e.  om  /\  B  C.  A )  -> 
( B  C.  A  ->  -.  A  ~~  B
) )
6766ex 423 . . 3  |-  ( A  e.  om  ->  ( B  C.  A  ->  ( B  C.  A  ->  -.  A  ~~  B ) ) )
6867pm2.43d 44 . 2  |-  ( A  e.  om  ->  ( B  C.  A  ->  -.  A  ~~  B ) )
6968imp 418 1  |-  ( ( A  e.  om  /\  B  C.  A )  ->  -.  A  ~~  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165    C. wpss 3166   (/)c0 3468   {csn 3653   class class class wbr 4039   Ord word 4407   suc csuc 4410   omcom 4672    ~~ cen 6876    ~<_ cdom 6877
This theorem is referenced by:  php2  7062  php3  7063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881
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