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Theorem phplem4 7043
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 6871 . 2  |-  ( suc 
A  ~~  suc  B  <->  E. f 
f : suc  A -1-1-onto-> suc  B )
2 f1of1 5471 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -1-1-> suc 
B )
32adantl 452 . . . . . . . . 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 4602 . . . . . . . . 9  |-  suc  B  e.  _V
6 sssucid 4469 . . . . . . . . . 10  |-  A  C_  suc  A
7 phplem2.1 . . . . . . . . . 10  |-  A  e. 
_V
8 f1imaen2g 6922 . . . . . . . . . 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 666 . . . . . . . . 9  |-  ( ( f : suc  A -1-1-> suc 
B  /\  suc  B  e. 
_V )  ->  (
f " A ) 
~~  A )
103, 5, 9sylancl 643 . . . . . . . 8  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( f " A )  ~~  A
)
11 ensym 6910 . . . . . . . 8  |-  ( ( f " A ) 
~~  A  ->  A  ~~  ( f " A
) )
1210, 11syl 15 . . . . . . 7  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  A  ~~  ( f " A
) )
13 nnord 4664 . . . . . . . . . 10  |-  ( A  e.  om  ->  Ord  A )
14 orddif 4486 . . . . . . . . . 10  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
1513, 14syl 15 . . . . . . . . 9  |-  ( A  e.  om  ->  A  =  ( suc  A  \  { A } ) )
1615imaeq2d 5012 . . . . . . . 8  |-  ( A  e.  om  ->  (
f " A )  =  ( f "
( suc  A  \  { A } ) ) )
17 f1ofn 5473 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f  Fn  suc  A
)
187sucid 4471 . . . . . . . . . . 11  |-  A  e. 
suc  A
19 fnsnfv 5582 . . . . . . . . . . 11  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  { (
f `  A ) }  =  ( f " { A } ) )
2017, 18, 19sylancl 643 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  { ( f `  A ) }  =  ( f " { A } ) )
2120difeq2d 3294 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( ( f " suc  A )  \  {
( f `  A
) } )  =  ( ( f " suc  A )  \  (
f " { A } ) ) )
22 imadmrn 5024 . . . . . . . . . . . 12  |-  ( f
" dom  f )  =  ran  f
2322eqcomi 2287 . . . . . . . . . . 11  |-  ran  f  =  ( f " dom  f )
24 f1ofo 5479 . . . . . . . . . . . 12  |-  ( f : suc  A -1-1-onto-> suc  B  ->  f : suc  A -onto-> suc  B )
25 forn 5454 . . . . . . . . . . . 12  |-  ( f : suc  A -onto-> suc  B  ->  ran  f  =  suc  B )
2624, 25syl 15 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ran  f  =  suc  B )
27 f1odm 5476 . . . . . . . . . . . 12  |-  ( f : suc  A -1-1-onto-> suc  B  ->  dom  f  =  suc  A )
2827imaeq2d 5012 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " dom  f )  =  ( f " suc  A
) )
2923, 26, 283eqtr3a 2339 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  suc  B  =  ( f " suc  A
) )
3029difeq1d 3293 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( suc  B  \  { ( f `  A ) } )  =  ( ( f
" suc  A )  \  { ( f `  A ) } ) )
31 dff1o3 5478 . . . . . . . . . . 11  |-  ( f : suc  A -1-1-onto-> suc  B  <->  ( f : suc  A -onto-> suc  B  /\  Fun  `' f ) )
3231simprbi 450 . . . . . . . . . 10  |-  ( f : suc  A -1-1-onto-> suc  B  ->  Fun  `' f )
33 imadif 5327 . . . . . . . . . 10  |-  ( Fun  `' f  ->  ( f
" ( suc  A  \  { A } ) )  =  ( ( f " suc  A
)  \  ( f " { A } ) ) )
3432, 33syl 15 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " ( suc  A  \  { A } ) )  =  ( ( f " suc  A )  \  (
f " { A } ) ) )
3521, 30, 343eqtr4rd 2326 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f " ( suc  A  \  { A } ) )  =  ( suc  B  \  { ( f `  A ) } ) )
3616, 35sylan9eq 2335 . . . . . . 7  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( f " A )  =  ( suc  B  \  {
( f `  A
) } ) )
3712, 36breqtrd 4047 . . . . . 6  |-  ( ( A  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  A  ~~  ( suc  B  \  {
( f `  A
) } ) )
38 fnfvelrn 5662 . . . . . . . . . 10  |-  ( ( f  Fn  suc  A  /\  A  e.  suc  A )  ->  ( f `  A )  e.  ran  f )
3917, 18, 38sylancl 643 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f `  A
)  e.  ran  f
)
4025eleq2d 2350 . . . . . . . . . 10  |-  ( f : suc  A -onto-> suc  B  ->  ( ( f `
 A )  e. 
ran  f  <->  ( f `  A )  e.  suc  B ) )
4124, 40syl 15 . . . . . . . . 9  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( ( f `  A )  e.  ran  f 
<->  ( f `  A
)  e.  suc  B
) )
4239, 41mpbid 201 . . . . . . . 8  |-  ( f : suc  A -1-1-onto-> suc  B  ->  ( f `  A
)  e.  suc  B
)
43 fvex 5539 . . . . . . . . 9  |-  ( f `
 A )  e. 
_V
444, 43phplem3 7042 . . . . . . . 8  |-  ( ( B  e.  om  /\  ( f `  A
)  e.  suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
4542, 44sylan2 460 . . . . . . 7  |-  ( ( B  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  B  ~~  ( suc  B  \  {
( f `  A
) } ) )
46 ensym 6910 . . . . . . 7  |-  ( B 
~~  ( suc  B  \  { ( f `  A ) } )  ->  ( suc  B  \  { ( f `  A ) } ) 
~~  B )
4745, 46syl 15 . . . . . 6  |-  ( ( B  e.  om  /\  f : suc  A -1-1-onto-> suc  B
)  ->  ( suc  B 
\  { ( f `
 A ) } )  ~~  B )
48 entr 6913 . . . . . 6  |-  ( ( A  ~~  ( suc 
B  \  { (
f `  A ) } )  /\  ( suc  B  \  { ( f `  A ) } )  ~~  B
)  ->  A  ~~  B )
4937, 47, 48syl2an 463 . . . . 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
)
5049anandirs 804 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  f : suc  A -1-1-onto-> suc  B )  ->  A  ~~  B )
5150ex 423 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( f : suc  A -1-1-onto-> suc 
B  ->  A  ~~  B ) )
5251exlimdv 1664 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( E. f  f : suc  A -1-1-onto-> suc  B  ->  A  ~~  B ) )
531, 52syl5bi 208 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  ~~  suc  B  ->  A  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023   Ord word 4391   suc csuc 4394   omcom 4656   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255    ~~ cen 6860
This theorem is referenced by:  nneneq  7044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864
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