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Theorem phplem4 7701
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 7527 . 2  |-  ( suc 
A  ~~  suc  B  <->  E. f 
f : suc  A -1-1-onto-> suc  B )
2 f1of1 5805 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -1-1-> suc 
B )
32adantl 466 . . . . . . . . 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 6631 . . . . . . . . 9  |-  suc  B  e.  _V
6 sssucid 4945 . . . . . . . . . 10  |-  A  C_  suc  A
7 phplem2.1 . . . . . . . . . 10  |-  A  e. 
_V
8 f1imaen2g 7578 . . . . . . . . . 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 685 . . . . . . . . 9  |-  ( ( f : suc  A -1-1-> suc 
B  /\  suc  B  e. 
_V )  ->  (
f " A ) 
~~  A )
103, 5, 9sylancl 662 . . . . . . . 8  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( f " A )  ~~  A
)
1110ensymd 7568 . . . . . . 7  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  A  ~~  ( f " A
) )
12 nnord 6693 . . . . . . . . . 10  |-  ( A  e.  om  ->  Ord  A )
13 orddif 4961 . . . . . . . . . 10  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
1412, 13syl 16 . . . . . . . . 9  |-  ( A  e.  om  ->  A  =  ( suc  A  \  { A } ) )
1514imaeq2d 5327 . . . . . . . 8  |-  ( A  e.  om  ->  (
f " A )  =  ( f "
( suc  A  \  { A } ) ) )
16 f1ofn 5807 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f  Fn  suc  A
)
177sucid 4947 . . . . . . . . . . 11  |-  A  e. 
suc  A
18 fnsnfv 5918 . . . . . . . . . . 11  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  { (
f `  A ) }  =  ( f " { A } ) )
1916, 17, 18sylancl 662 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  { ( f `  A ) }  =  ( f " { A } ) )
2019difeq2d 3607 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( ( f " suc  A )  \  {
( f `  A
) } )  =  ( ( f " suc  A )  \  (
f " { A } ) ) )
21 imadmrn 5337 . . . . . . . . . . . 12  |-  ( f
" dom  f )  =  ran  f
2221eqcomi 2456 . . . . . . . . . . 11  |-  ran  f  =  ( f " dom  f )
23 f1ofo 5813 . . . . . . . . . . . 12  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -onto-> suc  B )
24 forn 5788 . . . . . . . . . . . 12  |-  ( f : suc  A -onto-> suc  B  ->  ran  f  =  suc  B )
2523, 24syl 16 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ran  f  =  suc  B )
26 f1odm 5810 . . . . . . . . . . . 12  |-  ( f : suc  A -1-1-onto-> suc  B  ->  dom  f  =  suc  A )
2726imaeq2d 5327 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " dom  f )  =  ( f " suc  A
) )
2822, 25, 273eqtr3a 2508 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  suc  B  =  ( f " suc  A
) )
2928difeq1d 3606 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( suc  B  \  { ( f `  A ) } )  =  ( ( f
" suc  A )  \  { ( f `  A ) } ) )
30 dff1o3 5812 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  <->  ( f : suc  A -onto-> suc  B  /\  Fun  `' f ) )
3130simprbi 464 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  Fun  `' f )
32 imadif 5653 . . . . . . . . . 10  |-  ( Fun  `' f  ->  ( f
" ( suc  A  \  { A } ) )  =  ( ( f " suc  A
)  \  ( f " { A } ) ) )
3331, 32syl 16 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " ( suc  A  \  { A } ) )  =  ( ( f " suc  A )  \  (
f " { A } ) ) )
3420, 29, 333eqtr4rd 2495 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " ( suc  A  \  { A } ) )  =  ( suc  B  \  { ( f `  A ) } ) )
3515, 34sylan9eq 2504 . . . . . . 7  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( f " A )  =  ( suc  B  \  {
( f `  A
) } ) )
3611, 35breqtrd 4461 . . . . . 6  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  A  ~~  ( suc  B  \  {
( f `  A
) } ) )
37 fnfvelrn 6013 . . . . . . . . . 10  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  ( f `  A )  e.  ran  f )
3816, 17, 37sylancl 662 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f `  A
)  e.  ran  f
)
3924eleq2d 2513 . . . . . . . . . 10  |-  ( f : suc  A -onto-> suc  B  ->  ( ( f `
 A )  e. 
ran  f  <->  ( f `  A )  e.  suc  B ) )
4023, 39syl 16 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( ( f `  A )  e.  ran  f 
<->  ( f `  A
)  e.  suc  B
) )
4138, 40mpbid 210 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f `  A
)  e.  suc  B
)
42 fvex 5866 . . . . . . . . 9  |-  ( f `
 A )  e. 
_V
434, 42phplem3 7700 . . . . . . . 8  |-  ( ( B  e.  om  /\  ( f `  A
)  e.  suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
4441, 43sylan2 474 . . . . . . 7  |-  ( ( B  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
4544ensymd 7568 . . . . . 6  |-  ( ( B  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( suc  B 
\  { ( f `
 A ) } )  ~~  B )
46 entr 7569 . . . . . 6  |-  ( ( A  ~~  ( suc 
B  \  { (
f `  A ) } )  /\  ( suc  B  \  { ( f `  A ) } )  ~~  B
)  ->  A  ~~  B )
4736, 45, 46syl2an 477 . . . . 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 831 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  B )
4948ex 434 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( f : suc  A -1-1-onto-> suc 
B  ->  A  ~~  B ) )
5049exlimdv 1711 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( E. f  f : suc  A -1-1-onto-> suc  B  ->  A  ~~  B ) )
511, 50syl5bi 217 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  ~~  suc  B  ->  A  ~~  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804   _Vcvv 3095    \ cdif 3458    C_ wss 3461   {csn 4014   class class class wbr 4437   Ord word 4867   suc csuc 4870   `'ccnv 4988   dom cdm 4989   ran crn 4990   "cima 4992   Fun wfun 5572    Fn wfn 5573   -1-1->wf1 5575   -onto->wfo 5576   -1-1-onto->wf1o 5577   ` cfv 5578   omcom 6685    ~~ cen 7515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-er 7313  df-en 7519
This theorem is referenced by:  nneneq  7702
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