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Theorem phplem4 6854
Description: Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
Hypotheses
Ref Expression
phplem2.1  |-  A  e. 
_V
phplem2.2  |-  B  e. 
_V
Assertion
Ref Expression
phplem4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  ~~  suc  B  ->  A  ~~  B ) )

Proof of Theorem phplem4
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 bren 6746 . 2  |-  ( suc 
A  ~~  suc  B  <->  E. f 
f : suc  A -1-1-onto-> suc  B )
2 f1of1 5460 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -1-1-> suc 
B )
32adantl 277 . . . . . . . . 9  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  f : suc  A -1-1-> suc  B )
4 phplem2.2 . . . . . . . . . 10  |-  B  e. 
_V
54sucex 4498 . . . . . . . . 9  |-  suc  B  e.  _V
6 sssucid 4415 . . . . . . . . . 10  |-  A  C_  suc  A
7 phplem2.1 . . . . . . . . . 10  |-  A  e. 
_V
8 f1imaen2g 6792 . . . . . . . . . 10  |-  ( ( ( f : suc  A
-1-1-> suc  B  /\  suc  B  e.  _V )  /\  ( A  C_  suc  A  /\  A  e.  _V ) )  ->  (
f " A ) 
~~  A )
96, 7, 8mpanr12 439 . . . . . . . . 9  |-  ( ( f : suc  A -1-1-> suc 
B  /\  suc  B  e. 
_V )  ->  (
f " A ) 
~~  A )
103, 5, 9sylancl 413 . . . . . . . 8  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( f " A )  ~~  A
)
1110ensymd 6782 . . . . . . 7  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  A  ~~  ( f " A
) )
12 nnord 4611 . . . . . . . . . 10  |-  ( A  e.  om  ->  Ord  A )
13 orddif 4546 . . . . . . . . . 10  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
1412, 13syl 14 . . . . . . . . 9  |-  ( A  e.  om  ->  A  =  ( suc  A  \  { A } ) )
1514imaeq2d 4970 . . . . . . . 8  |-  ( A  e.  om  ->  (
f " A )  =  ( f "
( suc  A  \  { A } ) ) )
16 f1ofn 5462 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f  Fn  suc  A
)
177sucid 4417 . . . . . . . . . . 11  |-  A  e. 
suc  A
18 fnsnfv 5575 . . . . . . . . . . 11  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  { (
f `  A ) }  =  ( f " { A } ) )
1916, 17, 18sylancl 413 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  { ( f `  A ) }  =  ( f " { A } ) )
2019difeq2d 3253 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( ( f " suc  A )  \  {
( f `  A
) } )  =  ( ( f " suc  A )  \  (
f " { A } ) ) )
21 imadmrn 4980 . . . . . . . . . . . 12  |-  ( f
" dom  f )  =  ran  f
2221eqcomi 2181 . . . . . . . . . . 11  |-  ran  f  =  ( f " dom  f )
23 f1ofo 5468 . . . . . . . . . . . 12  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -onto-> suc  B )
24 forn 5441 . . . . . . . . . . . 12  |-  ( f : suc  A -onto-> suc  B  ->  ran  f  =  suc  B )
2523, 24syl 14 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ran  f  =  suc  B )
26 f1odm 5465 . . . . . . . . . . . 12  |-  ( f : suc  A -1-1-onto-> suc  B  ->  dom  f  =  suc  A )
2726imaeq2d 4970 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " dom  f )  =  ( f " suc  A
) )
2822, 25, 273eqtr3a 2234 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  suc  B  =  ( f " suc  A
) )
2928difeq1d 3252 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( suc  B  \  { ( f `  A ) } )  =  ( ( f
" suc  A )  \  { ( f `  A ) } ) )
30 dff1o3 5467 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  <->  ( f : suc  A -onto-> suc  B  /\  Fun  `' f ) )
3130simprbi 275 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  Fun  `' f )
32 imadif 5296 . . . . . . . . . 10  |-  ( Fun  `' f  ->  ( f
" ( suc  A  \  { A } ) )  =  ( ( f " suc  A
)  \  ( f " { A } ) ) )
3331, 32syl 14 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " ( suc  A  \  { A } ) )  =  ( ( f " suc  A )  \  (
f " { A } ) ) )
3420, 29, 333eqtr4rd 2221 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " ( suc  A  \  { A } ) )  =  ( suc  B  \  { ( f `  A ) } ) )
3515, 34sylan9eq 2230 . . . . . . 7  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( f " A )  =  ( suc  B  \  {
( f `  A
) } ) )
3611, 35breqtrd 4029 . . . . . 6  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  A  ~~  ( suc  B  \  {
( f `  A
) } ) )
37 fnfvelrn 5648 . . . . . . . . . 10  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  ( f `  A )  e.  ran  f )
3816, 17, 37sylancl 413 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f `  A
)  e.  ran  f
)
3924eleq2d 2247 . . . . . . . . . 10  |-  ( f : suc  A -onto-> suc  B  ->  ( ( f `
 A )  e. 
ran  f  <->  ( f `  A )  e.  suc  B ) )
4023, 39syl 14 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( ( f `  A )  e.  ran  f 
<->  ( f `  A
)  e.  suc  B
) )
4138, 40mpbid 147 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f `  A
)  e.  suc  B
)
42 vex 2740 . . . . . . . . . 10  |-  f  e. 
_V
4342, 7fvex 5535 . . . . . . . . 9  |-  ( f `
 A )  e. 
_V
444, 43phplem3 6853 . . . . . . . 8  |-  ( ( B  e.  om  /\  ( f `  A
)  e.  suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
4541, 44sylan2 286 . . . . . . 7  |-  ( ( B  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
4645ensymd 6782 . . . . . 6  |-  ( ( B  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( suc  B 
\  { ( f `
 A ) } )  ~~  B )
47 entr 6783 . . . . . 6  |-  ( ( A  ~~  ( suc 
B  \  { (
f `  A ) } )  /\  ( suc  B  \  { ( f `  A ) } )  ~~  B
)  ->  A  ~~  B )
4836, 46, 47syl2an 289 . . . . 5  |-  ( ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B )  /\  ( B  e.  om  /\  f : suc  A -1-1-onto-> suc  B ) )  ->  A  ~~  B
)
4948anandirs 593 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  B )
5049ex 115 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( f : suc  A -1-1-onto-> suc 
B  ->  A  ~~  B ) )
5150exlimdv 1819 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( E. f  f : suc  A -1-1-onto-> suc  B  ->  A  ~~  B ) )
521, 51biimtrid 152 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  ~~  suc  B  ->  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148   _Vcvv 2737    \ cdif 3126    C_ wss 3129   {csn 3592   class class class wbr 4003   Ord word 4362   suc csuc 4365   omcom 4589   `'ccnv 4625   dom cdm 4626   ran crn 4627   "cima 4629   Fun wfun 5210    Fn wfn 5211   -1-1->wf1 5213   -onto->wfo 5214   -1-1-onto->wf1o 5215   ` cfv 5216    ~~ cen 6737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-er 6534  df-en 6740
This theorem is referenced by:  nneneq  6856  php5  6857
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