MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infdiffi Unicode version

Theorem infdiffi 7358
Description: Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
infdiffi  |-  ( ( om  ~<_  A  /\  B  e.  Fin )  ->  ( A  \  B )  ~~  A )

Proof of Theorem infdiffi
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difeq2 3288 . . . . . 6  |-  ( x  =  (/)  ->  ( A 
\  x )  =  ( A  \  (/) ) )
2 dif0 3524 . . . . . 6  |-  ( A 
\  (/) )  =  A
31, 2syl6eq 2331 . . . . 5  |-  ( x  =  (/)  ->  ( A 
\  x )  =  A )
43breq1d 4033 . . . 4  |-  ( x  =  (/)  ->  ( ( A  \  x ) 
~~  A  <->  A  ~~  A ) )
54imbi2d 307 . . 3  |-  ( x  =  (/)  ->  ( ( om  ~<_  A  ->  ( A  \  x )  ~~  A )  <->  ( om  ~<_  A  ->  A  ~~  A
) ) )
6 difeq2 3288 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
76breq1d 4033 . . . 4  |-  ( x  =  y  ->  (
( A  \  x
)  ~~  A  <->  ( A  \  y )  ~~  A
) )
87imbi2d 307 . . 3  |-  ( x  =  y  ->  (
( om  ~<_  A  -> 
( A  \  x
)  ~~  A )  <->  ( om  ~<_  A  ->  ( A  \  y )  ~~  A ) ) )
9 difeq2 3288 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( A  \  x )  =  ( A  \  ( y  u.  { z } ) ) )
10 difun1 3428 . . . . . 6  |-  ( A 
\  ( y  u. 
{ z } ) )  =  ( ( A  \  y ) 
\  { z } )
119, 10syl6eq 2331 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( A  \  x )  =  ( ( A  \  y
)  \  { z } ) )
1211breq1d 4033 . . . 4  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( A 
\  x )  ~~  A 
<->  ( ( A  \ 
y )  \  {
z } )  ~~  A ) )
1312imbi2d 307 . . 3  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( om  ~<_  A  ->  ( A  \  x )  ~~  A
)  <->  ( om  ~<_  A  -> 
( ( A  \ 
y )  \  {
z } )  ~~  A ) ) )
14 difeq2 3288 . . . . 5  |-  ( x  =  B  ->  ( A  \  x )  =  ( A  \  B
) )
1514breq1d 4033 . . . 4  |-  ( x  =  B  ->  (
( A  \  x
)  ~~  A  <->  ( A  \  B )  ~~  A
) )
1615imbi2d 307 . . 3  |-  ( x  =  B  ->  (
( om  ~<_  A  -> 
( A  \  x
)  ~~  A )  <->  ( om  ~<_  A  ->  ( A  \  B )  ~~  A ) ) )
17 reldom 6869 . . . . 5  |-  Rel  ~<_
1817brrelex2i 4730 . . . 4  |-  ( om  ~<_  A  ->  A  e.  _V )
19 enrefg 6893 . . . 4  |-  ( A  e.  _V  ->  A  ~~  A )
2018, 19syl 15 . . 3  |-  ( om  ~<_  A  ->  A  ~~  A )
21 domen2 7004 . . . . . . . . 9  |-  ( ( A  \  y ) 
~~  A  ->  ( om 
~<_  ( A  \  y
)  <->  om  ~<_  A ) )
2221biimparc 473 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  om  ~<_  ( A 
\  y ) )
23 infdifsn 7357 . . . . . . . 8  |-  ( om  ~<_  ( A  \  y
)  ->  ( ( A  \  y )  \  { z } ) 
~~  ( A  \ 
y ) )
2422, 23syl 15 . . . . . . 7  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  (
( A  \  y
)  \  { z } )  ~~  ( A  \  y ) )
25 entr 6913 . . . . . . 7  |-  ( ( ( ( A  \ 
y )  \  {
z } )  ~~  ( A  \  y
)  /\  ( A  \  y )  ~~  A
)  ->  ( ( A  \  y )  \  { z } ) 
~~  A )
2624, 25sylancom 648 . . . . . 6  |-  ( ( om  ~<_  A  /\  ( A  \  y )  ~~  A )  ->  (
( A  \  y
)  \  { z } )  ~~  A
)
2726ex 423 . . . . 5  |-  ( om  ~<_  A  ->  ( ( A  \  y )  ~~  A  ->  ( ( A 
\  y )  \  { z } ) 
~~  A ) )
2827a2i 12 . . . 4  |-  ( ( om  ~<_  A  ->  ( A  \  y )  ~~  A )  ->  ( om 
~<_  A  ->  ( ( A  \  y )  \  { z } ) 
~~  A ) )
2928a1i 10 . . 3  |-  ( y  e.  Fin  ->  (
( om  ~<_  A  -> 
( A  \  y
)  ~~  A )  ->  ( om  ~<_  A  -> 
( ( A  \ 
y )  \  {
z } )  ~~  A ) ) )
305, 8, 13, 16, 20, 29findcard2 7098 . 2  |-  ( B  e.  Fin  ->  ( om 
~<_  A  ->  ( A  \  B )  ~~  A
) )
3130impcom 419 1  |-  ( ( om  ~<_  A  /\  B  e.  Fin )  ->  ( A  \  B )  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150   (/)c0 3455   {csn 3640   class class class wbr 4023   omcom 4656    ~~ cen 6860    ~<_ cdom 6861   Fincfn 6863
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-csb 3082  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-mpt 4079  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-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-fin 6867
  Copyright terms: Public domain W3C validator