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Theorem phplem4 7225
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 7053 . 2  |-  ( suc 
A  ~~  suc  B  <->  E. f 
f : suc  A -1-1-onto-> suc  B )
2 f1of1 5613 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -1-1-> suc 
B )
32adantl 453 . . . . . . . . 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 4731 . . . . . . . . 9  |-  suc  B  e.  _V
6 sssucid 4599 . . . . . . . . . 10  |-  A  C_  suc  A
7 phplem2.1 . . . . . . . . . 10  |-  A  e. 
_V
8 f1imaen2g 7104 . . . . . . . . . 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 667 . . . . . . . . 9  |-  ( ( f : suc  A -1-1-> suc 
B  /\  suc  B  e. 
_V )  ->  (
f " A ) 
~~  A )
103, 5, 9sylancl 644 . . . . . . . 8  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( f " A )  ~~  A
)
1110ensymd 7094 . . . . . . 7  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  A  ~~  ( f " A
) )
12 nnord 4793 . . . . . . . . . 10  |-  ( A  e.  om  ->  Ord  A )
13 orddif 4615 . . . . . . . . . 10  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
1412, 13syl 16 . . . . . . . . 9  |-  ( A  e.  om  ->  A  =  ( suc  A  \  { A } ) )
1514imaeq2d 5143 . . . . . . . 8  |-  ( A  e.  om  ->  (
f " A )  =  ( f "
( suc  A  \  { A } ) ) )
16 f1ofn 5615 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f  Fn  suc  A
)
177sucid 4601 . . . . . . . . . . 11  |-  A  e. 
suc  A
18 fnsnfv 5725 . . . . . . . . . . 11  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  { (
f `  A ) }  =  ( f " { A } ) )
1916, 17, 18sylancl 644 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  { ( f `  A ) }  =  ( f " { A } ) )
2019difeq2d 3408 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( ( f " suc  A )  \  {
( f `  A
) } )  =  ( ( f " suc  A )  \  (
f " { A } ) ) )
21 imadmrn 5155 . . . . . . . . . . . 12  |-  ( f
" dom  f )  =  ran  f
2221eqcomi 2391 . . . . . . . . . . 11  |-  ran  f  =  ( f " dom  f )
23 f1ofo 5621 . . . . . . . . . . . 12  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -onto-> suc  B )
24 forn 5596 . . . . . . . . . . . 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 5618 . . . . . . . . . . . 12  |-  ( f : suc  A -1-1-onto-> suc  B  ->  dom  f  =  suc  A )
2726imaeq2d 5143 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " dom  f )  =  ( f " suc  A
) )
2822, 25, 273eqtr3a 2443 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  suc  B  =  ( f " suc  A
) )
2928difeq1d 3407 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( suc  B  \  { ( f `  A ) } )  =  ( ( f
" suc  A )  \  { ( f `  A ) } ) )
30 dff1o3 5620 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  <->  ( f : suc  A -onto-> suc  B  /\  Fun  `' f ) )
3130simprbi 451 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  Fun  `' f )
32 imadif 5468 . . . . . . . . . 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 2430 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " ( suc  A  \  { A } ) )  =  ( suc  B  \  { ( f `  A ) } ) )
3515, 34sylan9eq 2439 . . . . . . 7  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( f " A )  =  ( suc  B  \  {
( f `  A
) } ) )
3611, 35breqtrd 4177 . . . . . 6  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  A  ~~  ( suc  B  \  {
( f `  A
) } ) )
37 fnfvelrn 5806 . . . . . . . . . 10  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  ( f `  A )  e.  ran  f )
3816, 17, 37sylancl 644 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f `  A
)  e.  ran  f
)
3924eleq2d 2454 . . . . . . . . . 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 202 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f `  A
)  e.  suc  B
)
42 fvex 5682 . . . . . . . . 9  |-  ( f `
 A )  e. 
_V
434, 42phplem3 7224 . . . . . . . 8  |-  ( ( B  e.  om  /\  ( f `  A
)  e.  suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
4441, 43sylan2 461 . . . . . . 7  |-  ( ( B  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
4544ensymd 7094 . . . . . 6  |-  ( ( B  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( suc  B 
\  { ( f `
 A ) } )  ~~  B )
46 entr 7095 . . . . . 6  |-  ( ( A  ~~  ( suc 
B  \  { (
f `  A ) } )  /\  ( suc  B  \  { ( f `  A ) } )  ~~  B
)  ->  A  ~~  B )
4736, 45, 46syl2an 464 . . . . 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 805 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  B )
4948ex 424 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( f : suc  A -1-1-onto-> suc 
B  ->  A  ~~  B ) )
5049exlimdv 1643 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( E. f  f : suc  A -1-1-onto-> suc  B  ->  A  ~~  B ) )
511, 50syl5bi 209 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  ~~  suc  B  ->  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2899    \ cdif 3260    C_ wss 3263   {csn 3757   class class class wbr 4153   Ord word 4521   suc csuc 4524   omcom 4785   `'ccnv 4817   dom cdm 4818   ran crn 4819   "cima 4821   Fun wfun 5388    Fn wfn 5389   -1-1->wf1 5391   -onto->wfo 5392   -1-1-onto->wf1o 5393   ` cfv 5394    ~~ cen 7042
This theorem is referenced by:  nneneq  7226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-er 6841  df-en 7046
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