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Theorem phpm 6752
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  E. x x  e.  ( A  \  B
) (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 6739 through phplem4 6742, nneneq 6744, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
Assertion
Ref Expression
phpm  |-  ( ( A  e.  om  /\  B  C_  A  /\  E. x  x  e.  ( A  \  B ) )  ->  -.  A  ~~  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem phpm
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  C_  A
)  /\  x  e.  ( A  \  B ) )  /\  A  =  (/) )  ->  A  =  (/) )
2 eldifi 3193 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  x  e.  A )
3 ne0i 3364 . . . . . . . . 9  |-  ( x  e.  A  ->  A  =/=  (/) )
42, 3syl 14 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  A  =/=  (/) )
54neneqd 2327 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  -.  A  =  (/) )
65ad2antlr 480 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  C_  A
)  /\  x  e.  ( A  \  B ) )  /\  A  =  (/) )  ->  -.  A  =  (/) )
71, 6pm2.21dd 609 . . . . 5  |-  ( ( ( ( A  e. 
om  /\  B  C_  A
)  /\  x  e.  ( A  \  B ) )  /\  A  =  (/) )  ->  -.  A  ~~  B )
8 php5dom 6750 . . . . . . . . . 10  |-  ( y  e.  om  ->  -.  suc  y  ~<_  y )
98ad2antlr 480 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  ->  -.  suc  y  ~<_  y )
10 simplr 519 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B ) )  /\  y  e.  om )  /\  A  =  suc  y )  /\  A  ~~  B )  ->  A  =  suc  y )
11 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B ) )  /\  y  e.  om )  /\  A  =  suc  y )  /\  A  ~~  B )  ->  A  ~~  B )
12 vex 2684 . . . . . . . . . . . . . . . 16  |-  y  e. 
_V
1312sucex 4410 . . . . . . . . . . . . . . 15  |-  suc  y  e.  _V
14 difss 3197 . . . . . . . . . . . . . . 15  |-  ( suc  y  \  { x } )  C_  suc  y
1513, 14ssexi 4061 . . . . . . . . . . . . . 14  |-  ( suc  y  \  { x } )  e.  _V
16 eldifn 3194 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( A  \  B )  ->  -.  x  e.  B )
1716ad3antlr 484 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  ->  -.  x  e.  B
)
18 simpllr 523 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e. 
om  /\  B  C_  A
)  /\  x  e.  ( A  \  B ) )  /\  y  e. 
om )  ->  B  C_  A )
1918adantr 274 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  ->  B  C_  A )
20 simpr 109 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  ->  A  =  suc  y )
2119, 20sseqtrd 3130 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  ->  B  C_  suc  y )
22 ssdif 3206 . . . . . . . . . . . . . . . 16  |-  ( B 
C_  suc  y  ->  ( B  \  { x } )  C_  ( suc  y  \  { x } ) )
23 disjsn 3580 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  i^i  { x } )  =  (/)  <->  -.  x  e.  B )
24 disj3 3410 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  i^i  { x } )  =  (/)  <->  B  =  ( B  \  { x } ) )
2523, 24bitr3i 185 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  e.  B  <->  B  =  ( B  \  { x } ) )
26 sseq1 3115 . . . . . . . . . . . . . . . . 17  |-  ( B  =  ( B  \  { x } )  ->  ( B  C_  ( suc  y  \  {
x } )  <->  ( B  \  { x } ) 
C_  ( suc  y  \  { x } ) ) )
2725, 26sylbi 120 . . . . . . . . . . . . . . . 16  |-  ( -.  x  e.  B  -> 
( B  C_  ( suc  y  \  { x } )  <->  ( B  \  { x } ) 
C_  ( suc  y  \  { x } ) ) )
2822, 27syl5ibr 155 . . . . . . . . . . . . . . 15  |-  ( -.  x  e.  B  -> 
( B  C_  suc  y  ->  B  C_  ( suc  y  \  { x } ) ) )
2917, 21, 28sylc 62 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  ->  B  C_  ( suc  y  \  { x } ) )
30 ssdomg 6665 . . . . . . . . . . . . . 14  |-  ( ( suc  y  \  {
x } )  e. 
_V  ->  ( B  C_  ( suc  y  \  {
x } )  ->  B  ~<_  ( suc  y  \  { x } ) ) )
3115, 29, 30mpsyl 65 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  ->  B  ~<_  ( suc  y  \  { x } ) )
32 simplr 519 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  -> 
y  e.  om )
332ad3antlr 484 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  ->  x  e.  A )
3433, 20eleqtrd 2216 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  ->  x  e.  suc  y )
35 phplem3g 6743 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  om  /\  x  e.  suc  y )  ->  y  ~~  ( suc  y  \  { x } ) )
3635ensymd 6670 . . . . . . . . . . . . . 14  |-  ( ( y  e.  om  /\  x  e.  suc  y )  ->  ( suc  y  \  { x } ) 
~~  y )
3732, 34, 36syl2anc 408 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  -> 
( suc  y  \  { x } ) 
~~  y )
38 domentr 6678 . . . . . . . . . . . . 13  |-  ( ( B  ~<_  ( suc  y  \  { x } )  /\  ( suc  y  \  { x } ) 
~~  y )  ->  B  ~<_  y )
3931, 37, 38syl2anc 408 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  ->  B  ~<_  y )
4039adantr 274 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B ) )  /\  y  e.  om )  /\  A  =  suc  y )  /\  A  ~~  B )  ->  B  ~<_  y )
41 endomtr 6677 . . . . . . . . . . 11  |-  ( ( A  ~~  B  /\  B  ~<_  y )  ->  A  ~<_  y )
4211, 40, 41syl2anc 408 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B ) )  /\  y  e.  om )  /\  A  =  suc  y )  /\  A  ~~  B )  ->  A  ~<_  y )
4310, 42eqbrtrrd 3947 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B ) )  /\  y  e.  om )  /\  A  =  suc  y )  /\  A  ~~  B )  ->  suc  y  ~<_  y )
449, 43mtand 654 . . . . . . . 8  |-  ( ( ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B
) )  /\  y  e.  om )  /\  A  =  suc  y )  ->  -.  A  ~~  B )
4544ex 114 . . . . . . 7  |-  ( ( ( ( A  e. 
om  /\  B  C_  A
)  /\  x  e.  ( A  \  B ) )  /\  y  e. 
om )  ->  ( A  =  suc  y  ->  -.  A  ~~  B ) )
4645rexlimdva 2547 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B ) )  ->  ( E. y  e.  om  A  =  suc  y  ->  -.  A  ~~  B ) )
4746imp 123 . . . . 5  |-  ( ( ( ( A  e. 
om  /\  B  C_  A
)  /\  x  e.  ( A  \  B ) )  /\  E. y  e.  om  A  =  suc  y )  ->  -.  A  ~~  B )
48 nn0suc 4513 . . . . . 6  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. y  e.  om  A  =  suc  y ) )
4948ad2antrr 479 . . . . 5  |-  ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B ) )  ->  ( A  =  (/)  \/  E. y  e. 
om  A  =  suc  y ) )
507, 47, 49mpjaodan 787 . . . 4  |-  ( ( ( A  e.  om  /\  B  C_  A )  /\  x  e.  ( A  \  B ) )  ->  -.  A  ~~  B )
5150ex 114 . . 3  |-  ( ( A  e.  om  /\  B  C_  A )  -> 
( x  e.  ( A  \  B )  ->  -.  A  ~~  B ) )
5251exlimdv 1791 . 2  |-  ( ( A  e.  om  /\  B  C_  A )  -> 
( E. x  x  e.  ( A  \  B )  ->  -.  A  ~~  B ) )
53523impia 1178 1  |-  ( ( A  e.  om  /\  B  C_  A  /\  E. x  x  e.  ( A  \  B ) )  ->  -.  A  ~~  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480    =/= wne 2306   E.wrex 2415   _Vcvv 2681    \ cdif 3063    i^i cin 3065    C_ wss 3066   (/)c0 3358   {csn 3522   class class class wbr 3924   suc csuc 4282   omcom 4499    ~~ cen 6625    ~<_ cdom 6626
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 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-er 6422  df-en 6628  df-dom 6629
This theorem is referenced by:  phpelm  6753
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