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Theorem phplem4 7763
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 7589 . 2  |-  ( suc 
A  ~~  suc  B  <->  E. f 
f : suc  A -1-1-onto-> suc  B )
2 f1of1 5830 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -1-1-> suc 
B )
32adantl 467 . . . . . . . . 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 6652 . . . . . . . . 9  |-  suc  B  e.  _V
6 sssucid 5519 . . . . . . . . . 10  |-  A  C_  suc  A
7 phplem2.1 . . . . . . . . . 10  |-  A  e. 
_V
8 f1imaen2g 7640 . . . . . . . . . 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 689 . . . . . . . . 9  |-  ( ( f : suc  A -1-1-> suc 
B  /\  suc  B  e. 
_V )  ->  (
f " A ) 
~~  A )
103, 5, 9sylancl 666 . . . . . . . 8  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( f " A )  ~~  A
)
1110ensymd 7630 . . . . . . 7  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  A  ~~  ( f " A
) )
12 nnord 6714 . . . . . . . . . 10  |-  ( A  e.  om  ->  Ord  A )
13 orddif 5535 . . . . . . . . . 10  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
1412, 13syl 17 . . . . . . . . 9  |-  ( A  e.  om  ->  A  =  ( suc  A  \  { A } ) )
1514imaeq2d 5187 . . . . . . . 8  |-  ( A  e.  om  ->  (
f " A )  =  ( f "
( suc  A  \  { A } ) ) )
16 f1ofn 5832 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f  Fn  suc  A
)
177sucid 5521 . . . . . . . . . . 11  |-  A  e. 
suc  A
18 fnsnfv 5941 . . . . . . . . . . 11  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  { (
f `  A ) }  =  ( f " { A } ) )
1916, 17, 18sylancl 666 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  { ( f `  A ) }  =  ( f " { A } ) )
2019difeq2d 3583 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( ( f " suc  A )  \  {
( f `  A
) } )  =  ( ( f " suc  A )  \  (
f " { A } ) ) )
21 imadmrn 5197 . . . . . . . . . . . 12  |-  ( f
" dom  f )  =  ran  f
2221eqcomi 2435 . . . . . . . . . . 11  |-  ran  f  =  ( f " dom  f )
23 f1ofo 5838 . . . . . . . . . . . 12  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -onto-> suc  B )
24 forn 5813 . . . . . . . . . . . 12  |-  ( f : suc  A -onto-> suc  B  ->  ran  f  =  suc  B )
2523, 24syl 17 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ran  f  =  suc  B )
26 f1odm 5835 . . . . . . . . . . . 12  |-  ( f : suc  A -1-1-onto-> suc  B  ->  dom  f  =  suc  A )
2726imaeq2d 5187 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " dom  f )  =  ( f " suc  A
) )
2822, 25, 273eqtr3a 2487 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  suc  B  =  ( f " suc  A
) )
2928difeq1d 3582 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( suc  B  \  { ( f `  A ) } )  =  ( ( f
" suc  A )  \  { ( f `  A ) } ) )
30 dff1o3 5837 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  <->  ( f : suc  A -onto-> suc  B  /\  Fun  `' f ) )
3130simprbi 465 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  Fun  `' f )
32 imadif 5676 . . . . . . . . . 10  |-  ( Fun  `' f  ->  ( f
" ( suc  A  \  { A } ) )  =  ( ( f " suc  A
)  \  ( f " { A } ) ) )
3331, 32syl 17 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " ( suc  A  \  { A } ) )  =  ( ( f " suc  A )  \  (
f " { A } ) ) )
3420, 29, 333eqtr4rd 2474 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " ( suc  A  \  { A } ) )  =  ( suc  B  \  { ( f `  A ) } ) )
3515, 34sylan9eq 2483 . . . . . . 7  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( f " A )  =  ( suc  B  \  {
( f `  A
) } ) )
3611, 35breqtrd 4448 . . . . . 6  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  A  ~~  ( suc  B  \  {
( f `  A
) } ) )
37 fnfvelrn 6034 . . . . . . . . . 10  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  ( f `  A )  e.  ran  f )
3816, 17, 37sylancl 666 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f `  A
)  e.  ran  f
)
3924eleq2d 2492 . . . . . . . . . 10  |-  ( f : suc  A -onto-> suc  B  ->  ( ( f `
 A )  e. 
ran  f  <->  ( f `  A )  e.  suc  B ) )
4023, 39syl 17 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( ( f `  A )  e.  ran  f 
<->  ( f `  A
)  e.  suc  B
) )
4138, 40mpbid 213 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f `  A
)  e.  suc  B
)
42 fvex 5891 . . . . . . . . 9  |-  ( f `
 A )  e. 
_V
434, 42phplem3 7762 . . . . . . . 8  |-  ( ( B  e.  om  /\  ( f `  A
)  e.  suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
4441, 43sylan2 476 . . . . . . 7  |-  ( ( B  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
4544ensymd 7630 . . . . . 6  |-  ( ( B  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( suc  B 
\  { ( f `
 A ) } )  ~~  B )
46 entr 7631 . . . . . 6  |-  ( ( A  ~~  ( suc 
B  \  { (
f `  A ) } )  /\  ( suc  B  \  { ( f `  A ) } )  ~~  B
)  ->  A  ~~  B )
4736, 45, 46syl2an 479 . . . . 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
)
4847anandirs 838 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  B )
4948ex 435 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( f : suc  A -1-1-onto-> suc 
B  ->  A  ~~  B ) )
5049exlimdv 1772 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( E. f  f : suc  A -1-1-onto-> suc  B  ->  A  ~~  B ) )
511, 50syl5bi 220 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  ~~  suc  B  ->  A  ~~  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   _Vcvv 3080    \ cdif 3433    C_ wss 3436   {csn 3998   class class class wbr 4423   `'ccnv 4852   dom cdm 4853   ran crn 4854   "cima 4856   Ord word 5441   suc csuc 5444   Fun wfun 5595    Fn wfn 5596   -1-1->wf1 5598   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601   omcom 6706    ~~ cen 7577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-er 7374  df-en 7581
This theorem is referenced by:  nneneq  7764
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