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Theorem php 7013
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 7008 through phplem4 7011, nneneq 7012, 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
StepHypRef Expression
1 0ss 3458 . . . . . . . 8  |-  (/)  C_  B
2 sspsstr 3256 . . . . . . . 8  |-  ( (
(/)  C_  B  /\  B  C.  A )  ->  (/)  C.  A
)
31, 2mpan 654 . . . . . . 7  |-  ( B 
C.  A  ->  (/)  C.  A
)
4 0pss 3467 . . . . . . . 8  |-  ( (/)  C.  A  <->  A  =/=  (/) )
5 df-ne 2423 . . . . . . . 8  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
64, 5bitri 242 . . . . . . 7  |-  ( (/)  C.  A  <->  -.  A  =  (/) )
73, 6sylib 190 . . . . . 6  |-  ( B 
C.  A  ->  -.  A  =  (/) )
8 nn0suc 4652 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
98orcanai 884 . . . . . 6  |-  ( ( A  e.  om  /\  -.  A  =  (/) )  ->  E. x  e.  om  A  =  suc  x )
107, 9sylan2 462 . . . . 5  |-  ( ( A  e.  om  /\  B  C.  A )  ->  E. x  e.  om  A  =  suc  x )
11 pssnel 3494 . . . . . . . . . 10  |-  ( B 
C.  suc  x  ->  E. y ( y  e. 
suc  x  /\  -.  y  e.  B )
)
12 pssss 3246 . . . . . . . . . . . . . . . . 17  |-  ( B 
C.  suc  x  ->  B 
C_  suc  x )
13 ssdif 3286 . . . . . . . . . . . . . . . . . 18  |-  ( B 
C_  suc  x  ->  ( B  \  { y } )  C_  ( suc  x  \  { y } ) )
14 disjsn 3667 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  i^i  { y } )  =  (/)  <->  -.  y  e.  B )
15 disj3 3474 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  i^i  { y } )  =  (/)  <->  B  =  ( B  \  { y } ) )
1614, 15bitr3i 244 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  y  e.  B  <->  B  =  ( B  \  { y } ) )
17 sseq1 3174 . . . . . . . . . . . . . . . . . . 19  |-  ( B  =  ( B  \  { y } )  ->  ( B  C_  ( suc  x  \  {
y } )  <->  ( B  \  { y } ) 
C_  ( suc  x  \  { y } ) ) )
1816, 17sylbi 189 . . . . . . . . . . . . . . . . . 18  |-  ( -.  y  e.  B  -> 
( B  C_  ( suc  x  \  { y } )  <->  ( B  \  { y } ) 
C_  ( suc  x  \  { y } ) ) )
1913, 18syl5ibr 214 . . . . . . . . . . . . . . . . 17  |-  ( -.  y  e.  B  -> 
( B  C_  suc  x  ->  B  C_  ( suc  x  \  { y } ) ) )
20 vex 2766 . . . . . . . . . . . . . . . . . . . 20  |-  x  e. 
_V
2120sucex 4574 . . . . . . . . . . . . . . . . . . 19  |-  suc  x  e.  _V
22 difss 3278 . . . . . . . . . . . . . . . . . . 19  |-  ( suc  x  \  { y } )  C_  suc  x
2321, 22ssexi 4133 . . . . . . . . . . . . . . . . . 18  |-  ( suc  x  \  { y } )  e.  _V
24 ssdomg 6875 . . . . . . . . . . . . . . . . . 18  |-  ( ( suc  x  \  {
y } )  e. 
_V  ->  ( B  C_  ( suc  x  \  {
y } )  ->  B  ~<_  ( suc  x  \  { y } ) ) )
2523, 24ax-mp 10 . . . . . . . . . . . . . . . . 17  |-  ( B 
C_  ( suc  x  \  { y } )  ->  B  ~<_  ( suc  x  \  { y } ) )
2612, 19, 25syl56 32 . . . . . . . . . . . . . . . 16  |-  ( -.  y  e.  B  -> 
( B  C.  suc  x  ->  B  ~<_  ( suc  x  \  { y } ) ) )
2726imp 420 . . . . . . . . . . . . . . 15  |-  ( ( -.  y  e.  B  /\  B  C.  suc  x
)  ->  B  ~<_  ( suc  x  \  { y } ) )
28 vex 2766 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
2920, 28phplem3 7010 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  om  /\  y  e.  suc  x )  ->  x  ~~  ( suc  x  \  { y } ) )
30 ensym 6878 . . . . . . . . . . . . . . . 16  |-  ( x 
~~  ( suc  x  \  { y } )  ->  ( suc  x  \  { y } ) 
~~  x )
3129, 30syl 17 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  om  /\  y  e.  suc  x )  ->  ( suc  x  \  { y } ) 
~~  x )
32 domentr 6888 . . . . . . . . . . . . . . 15  |-  ( ( B  ~<_  ( suc  x  \  { y } )  /\  ( suc  x  \  { y } ) 
~~  x )  ->  B  ~<_  x )
3327, 31, 32syl2an 465 . . . . . . . . . . . . . 14  |-  ( ( ( -.  y  e.  B  /\  B  C.  suc  x )  /\  (
x  e.  om  /\  y  e.  suc  x ) )  ->  B  ~<_  x )
3433exp43 598 . . . . . . . . . . . . 13  |-  ( -.  y  e.  B  -> 
( B  C.  suc  x  ->  ( x  e. 
om  ->  ( y  e. 
suc  x  ->  B  ~<_  x ) ) ) )
3534com4r 82 . . . . . . . . . . . 12  |-  ( y  e.  suc  x  -> 
( -.  y  e.  B  ->  ( B  C.  suc  x  ->  (
x  e.  om  ->  B  ~<_  x ) ) ) )
3635imp 420 . . . . . . . . . . 11  |-  ( ( y  e.  suc  x  /\  -.  y  e.  B
)  ->  ( B  C.  suc  x  ->  (
x  e.  om  ->  B  ~<_  x ) ) )
3736exlimiv 2024 . . . . . . . . . 10  |-  ( E. y ( y  e. 
suc  x  /\  -.  y  e.  B )  ->  ( B  C.  suc  x  ->  ( x  e. 
om  ->  B  ~<_  x ) ) )
3811, 37mpcom 34 . . . . . . . . 9  |-  ( B 
C.  suc  x  ->  ( x  e.  om  ->  B  ~<_  x ) )
39 endomtr 6887 . . . . . . . . . . . 12  |-  ( ( suc  x  ~~  B  /\  B  ~<_  x )  ->  suc  x  ~<_  x )
40 sssucid 4441 . . . . . . . . . . . . 13  |-  x  C_  suc  x
41 ssdomg 6875 . . . . . . . . . . . . 13  |-  ( suc  x  e.  _V  ->  ( x  C_  suc  x  ->  x  ~<_  suc  x )
)
4221, 40, 41mp2 19 . . . . . . . . . . . 12  |-  x  ~<_  suc  x
43 sbth 6949 . . . . . . . . . . . 12  |-  ( ( suc  x  ~<_  x  /\  x  ~<_  suc  x )  ->  suc  x  ~~  x
)
4439, 42, 43sylancl 646 . . . . . . . . . . 11  |-  ( ( suc  x  ~~  B  /\  B  ~<_  x )  ->  suc  x  ~~  x
)
4544expcom 426 . . . . . . . . . 10  |-  ( B  ~<_  x  ->  ( suc  x  ~~  B  ->  suc  x  ~~  x ) )
46 peano2b 4644 . . . . . . . . . . . . 13  |-  ( x  e.  om  <->  suc  x  e. 
om )
47 nnord 4636 . . . . . . . . . . . . 13  |-  ( suc  x  e.  om  ->  Ord 
suc  x )
4846, 47sylbi 189 . . . . . . . . . . . 12  |-  ( x  e.  om  ->  Ord  suc  x )
4920sucid 4443 . . . . . . . . . . . 12  |-  x  e. 
suc  x
50 nordeq 4383 . . . . . . . . . . . 12  |-  ( ( Ord  suc  x  /\  x  e.  suc  x )  ->  suc  x  =/=  x )
5148, 49, 50sylancl 646 . . . . . . . . . . 11  |-  ( x  e.  om  ->  suc  x  =/=  x )
52 nneneq 7012 . . . . . . . . . . . . . 14  |-  ( ( suc  x  e.  om  /\  x  e.  om )  ->  ( suc  x  ~~  x 
<->  suc  x  =  x ) )
5346, 52sylanb 460 . . . . . . . . . . . . 13  |-  ( ( x  e.  om  /\  x  e.  om )  ->  ( suc  x  ~~  x 
<->  suc  x  =  x ) )
5453anidms 629 . . . . . . . . . . . 12  |-  ( x  e.  om  ->  ( suc  x  ~~  x  <->  suc  x  =  x ) )
5554necon3bbid 2455 . . . . . . . . . . 11  |-  ( x  e.  om  ->  ( -.  suc  x  ~~  x  <->  suc  x  =/=  x ) )
5651, 55mpbird 225 . . . . . . . . . 10  |-  ( x  e.  om  ->  -.  suc  x  ~~  x )
5745, 56nsyli 135 . . . . . . . . 9  |-  ( B  ~<_  x  ->  ( x  e.  om  ->  -.  suc  x  ~~  B ) )
5838, 57syli 35 . . . . . . . 8  |-  ( B 
C.  suc  x  ->  ( x  e.  om  ->  -. 
suc  x  ~~  B
) )
5958com12 29 . . . . . . 7  |-  ( x  e.  om  ->  ( B  C.  suc  x  ->  -.  suc  x  ~~  B
) )
60 psseq2 3239 . . . . . . . 8  |-  ( A  =  suc  x  -> 
( B  C.  A  <->  B 
C.  suc  x )
)
61 breq1 4000 . . . . . . . . 9  |-  ( A  =  suc  x  -> 
( A  ~~  B  <->  suc  x  ~~  B ) )
6261notbid 287 . . . . . . . 8  |-  ( A  =  suc  x  -> 
( -.  A  ~~  B 
<->  -.  suc  x  ~~  B ) )
6360, 62imbi12d 313 . . . . . . 7  |-  ( A  =  suc  x  -> 
( ( B  C.  A  ->  -.  A  ~~  B )  <->  ( B  C.  suc  x  ->  -.  suc  x  ~~  B ) ) )
6459, 63syl5ibrcom 215 . . . . . 6  |-  ( x  e.  om  ->  ( A  =  suc  x  -> 
( B  C.  A  ->  -.  A  ~~  B
) ) )
6564rexlimiv 2636 . . . . 5  |-  ( E. x  e.  om  A  =  suc  x  ->  ( B  C.  A  ->  -.  A  ~~  B ) )
6610, 65syl 17 . . . 4  |-  ( ( A  e.  om  /\  B  C.  A )  -> 
( B  C.  A  ->  -.  A  ~~  B
) )
6766ex 425 . . 3  |-  ( A  e.  om  ->  ( B  C.  A  ->  ( B  C.  A  ->  -.  A  ~~  B ) ) )
6867pm2.43d 46 . 2  |-  ( A  e.  om  ->  ( B  C.  A  ->  -.  A  ~~  B ) )
6968imp 420 1  |-  ( ( A  e.  om  /\  B  C.  A )  ->  -.  A  ~~  B )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   _Vcvv 2763    \ cdif 3124    i^i cin 3126    C_ wss 3127    C. wpss 3128   (/)c0 3430   {csn 3614   class class class wbr 3997   Ord word 4363   suc csuc 4366   omcom 4628    ~~ cen 6828    ~<_ cdom 6829
This theorem is referenced by:  php2  7014  php3  7015
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-er 6628  df-en 6832  df-dom 6833
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